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j C.

A1 11 009901 08

ST5°5rvPl60;1977 C.1 NBS-PUB-C 1959

i/t Q NBS MONOGRAPH 160

Geometrical Considerations

and Nonmenclature for Reflectance

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

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janal Bureau ot Standardi

I NOV 8 1977

Geometrical Considerations

'•^"^^^ and Nomenclature for Reflectance

F.E. Nicodemus, J.C. Richmond,

and J.J. Hsia

Institute for Basic Standards

'

, National Bureau of Standards

i Washington, D.C. 20234

and

I.W. Ginsberg

EG&G, Inc. Las Vegas, Nevada 89101

and

T. Limperis

Agro Sciences, Inc. Ann Arbor, Michigan 48103

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

Dr. Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

(J , NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

issued October 1977
For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price $2.10

Stock No. 003-003-01793
Contents

Page

I. Introduction 1

II. Reflectance Concepts and Nomenclature 3

A. Bidirectional Reflectance-Distribution Function (BRDF) 3 B. Definitions of Reflectance and Reflectance Factor 7

C. Reflectances and Reflectance Factors for Nine Geometries 9

III. Some Additional Reflectance-Nomenclature Possibilities 12

A. McCamy's Notation for Right Circular Cones 12

B. Multivariate Reflectance-Distribution Functions (MRDF's) 14

C. Other Special Types (BRIDF and BRRDF) 19

IV. Practical Considerations for Measurement of Reflectance ^ 20

A. Effects of Finite Intervals 21

B. Definition of Reflecting-Surface Area 24

C. Sub-Surface Scattering (Edge Effects) 28

D. Extrapolation to Different Geometries 30

E. Other Radiation Parameters and Fluorescence (or Phosphorescence) __ 30

F. Use of Reference Standards 32

V. Summary, Conclusions, and Recommendations 33

Appendices

VI. A. Background and Basis for Recommendations 34

VII. B. Details of Reflectance Derivations (A Thought Experiment) 37

VIII. C. Perfectly Diffuse and Perfectly Specular Reflectances 43

IX. D. Details for Derivations of McCamy's Notation 46

X. Acknowledgments 50

XII. References 51

Illustrations

Figure

1. Geometry of incident and reflected beams (for general cases where subsurface scattering is involved) 4

2. Geometry of incident and reflected elementary beams (Z axis is chosen along

the normal to the surface element dA at O.) (Adapted from [10]) 6

3. Designations for denoting incident- and reflected- (collected-) beam geometry. _ 10

4. McCamy's notation for right circular cones 13

5. Three-dimensional graph of bidirectional reflectance-distribution function

(BRDF) /,(33.2°, 0°; e,</)) (from [10]) 22

6. Three configurations for physically defining the reflecting area. (See sec. IV.C concerning important limitations arising from sub-surface scattering or "edge effects".) 25

iii
Page

7. Simplified configuration when source (or receiver) is located in focal plane (or

just behind field stop in focal plane) of primary optics 26

8. Simplified configuration with source (or receiver) and sample (reflecting

surface) in conjugate image planes of primary optics 27

9. "Thought-experiment" configuration for bidirectional reflectance 38

10. "Thought-experiment" configuration for directional-conical reflectance 40

11. "Thought-experiment" configuration for conical-directional reflectance 41

12. Construction for deriving relationships between spherical coordinates and

McCamy's notation for right-circular cones 47

13. Details of the construction of figure 12 48

Tables

Table

1. Proposed nomenclature for nine kinds of reflectance 11

2. Proposed nomenclature for nine reflectance factors 12

3. The multivariate reflectance-distribution functions 15

4. Comparison between recommended reflectance nomenclature and that of [10] 35

iv
List of Abbreviations

Defined or used

in sec, ecj, Abbreviation 1 arm table, etc. BRDF bidirectional reflectance-distribution function II.A [eq (7)]. BRIDF bidirectional reflected-intensity-distribution function III.C [eq (26)].

BRRDF bidirectional reflected-radiance-distribution function III.C leq (25)].

BSSRDF bidirectional scattering-surface reflectance-distribution II.A [eqs (1) & (2)] function

MRDF multivariate reflectance-distribution function III.B [table 3]. RDF reflectance-distribution function III.C. SI Systeme International d'Unites (the International System of List of UnitUnits) Dimensions.

ORDF avariate reflectance-distribution function III.B [table 3]. IRDF univariate reflectance-distribution function III.B [table 3]. 2RDF bivariate reflectance-distribution function III.B [table 3]. 3RDF trivariate reflectance-distribution function III.B [table 3]. 4RDF quadrivariate reflectance-distribution function III.B [table 3].

List of Symbols

Defined or used

in sec, eq. Symbol Term Explanation Unit-Dimension table, fig., etc.

A area of a surface; = jjdx -dy [m^] II.A [eq (4)].

a- area of a band of diminishing [m^] IV.C [eq (34)]. reflected radiance; a(r,„) C constant with respect to wavelength [X] IV.E.

c speed of light electromagnetic radiation

[m-s-'] I (footnote).

in empty space; — (2.997

924 58 ± 0.000 000

012)xlO« [m s-'l E irradiance incident flux (surface) den- [W-m-2] II.A [eq (4)].

sity; = d<t>/dA

f frequency of modulation or fluctua- [Hz] I (footnote).

tion; f«v

/( ) function of the quantities or param- II.A [eq (9)]. eters in the parentheses

fAei,<t>i; BRDF bidirectional reflectance- [sr->] II.A [eq (6)]. distribution function

fAei,4>i; spectral BRDF spectral bidirectional re- [sr->] IV.E [eq (38)].

flectance-distribution

function

fr{><) relative spectral separable relative spectral [dimensionless] IV.E [eq (40)].

BRDF reflectance-distribution

function

V
List of Symbols (continued)

Defined or used

Term Explanation

m sec, eg, Unit-Dimension table, fig., etc.

average RDF for indicated

beam geometry

BRIDF

BRRDF

irradiance

radiant intensity

(of a source)

radiant intensity

radiance

direction cosine

radiant exitance

direction cosine

radiance

direction cosine

radiant flux (power)

reflectance fac- tor receiver

[see note at end of table 3, and accompanying text, concerning special rules

for designating parameter intervals for such av- eraging in the MRDF

scheme]

bidirectional reflected-in- tensity-distribution function bidirectional reflected-radiance-distribution function [obsolete—replaced hy E]

(now used for exposure

in [J m-2])

exitent flux per unit solid angle; = d^/du>

[obsolete—replaced by /] ^ d^(^/{dA -cos^-dw)

of cone axis with respect to /^-axis; = xJpo\ used in App. D only,

exitent flux (surface) density; = d*P/dA

of cone axis with respect to K-axis; = jo/Po; used in App. D only

[obsolete—replaced by L ] of cone axis with respect to Z-axis; = Zo/po? used in App. D only

[obsolete—replaced by 4>]

= d^M^rid

[designation used in figs. 9, 10, and 11, and accompanying text only]

[sr-'] III.B [table 3].

[sr-']

[sr-']

[W-m-2]

[W-sr-']

[W-sr-']

[Wm-=^-sr-']

[dimensionless]

[Wm-2]

[dimensionless]

[Wm-2sr-']

[dimensionless]

[W]

[dimensionless]

III.C [eq (26)].

III.C [eq (25)].

App. A [table 4].

III.C [eq (26)].

App. A [table 4].

II. A [eq (1)]. App. D [eq (D3)].

App. B [eq (B3)].

App. D [eq (D3)].

App. A [table 4]. App. D [eq (D3)].

App. A [table 4].

II. B. 2 [eq (18)].

App. B [figs. 9, 10,

and 11].

VI
List of Symbols (continued)

Defined or used

Symbol Term Explanation

m sec., eg, Unit-Dimension table, fig., etc.

width of reflect- ed beam

radial polar co- ordinate in X—Y plane

radius of base

of right circular cone

sub-surfacescattering dis- tance

partial reflec- tance maximum subsurface-scattering dist.

BSSRDF

width of ex- posed sample

surface

parameter; vari- able quantity

radiometric

quantity

width of reflected (and col- [m]

lected) beam in fig. 6

and accompanying text only

related to spherical coordi- [m]

nates by r = p sin^;

and to rectangular coor- dinates hy X = r-cos</),

y = r-sin(/) = p sinK in fig. 12 and [m]

related text only

separation between point [m]

of incidence and point of

significant reflection (exitance) on reference sur- face with sub-surface

scattering; = [{xr — Xj)^ + (yr -Yify.

used by De Vos [19] for [sr"']

the BRIDF/r/ of eq (26) maximum distance (across [m]

reference surface or nominal reflecting sur- face) for significant subsurface-scattering

bidirectional scattering-sur- [m"^

face reflectance-distribution function; —

S{di,4)i,Xi,yi;dr,(t)r,Xr,yr)

used in fig. 6 and accom- [m]

panying text only

[may also be a function of

other variables]

includes: ^ [J], (I>[W], [X]

/[W sr-'],£ or M[W m-2], or

IV. B [fig. 6].

iV.C [eq (32)].

App. D [fig. 12].

II. A [eqs (7) and

(8)].

III. C [following eq

(26)] only. IV. C [eq (32)].

•sr-'] II.A[eq(2)].

IV.B [fig. 6].

App. C [following

eq (CIO)]. IV. E [following eq

mi

L[W- •sr •]

vu
List of Symbols (continued)

Defined or used

Symbol

X',Y',Z'

X",Y",Z"

x,y,z

a

aid)

)8

)8

Term

rectangular-coordinate axes

"tilted" axes

"rotated" axes

relative spectroradiometric

quantity

rectangular position coordi- nates

off-specular

angle

limit function

radiance

factor

off-specular

angle

angle

Explanation

usually oriented with Zaxis normal to element

dA of reference surface

andA'— y axes in tangent

plane containing dA; Zaxis is also polar axis [6 = 0) for spherical coordi- nates p,d,4>.

[used in App. D only]

[used in App. D only] = Xk/C ; relative spectral

distribution of radiation

m sec, eg, Unit-Dimension table, fig., etc.

[nm-']

8( ) Dirac delta- function

X = distance from Y—Z [m]

plane, etc.

Or — di ± (X —direction in [rad]

which off-specular glints are observed

used to designate integra- [rad]

tion limits for cf) when

expressing reflectance

quantities in terms of the

BRDF in McCamy's notation for rt. -circular cones; = cos~'[(cosk —

cos $0 cos ^ )/(sin sin 0)] CIE symbol for a radiance

factor or • • -directional

reflectance factor R{ - ;6r,(t>r); also lumi- nance factor

(f>r — 4>i — —direction in [rad]

which off-specuJar glints are observed

angle between OP and O/o < [rad]

where P is any point on

a plane through Pq, per- pendicular to OPo ; = ^POP,

defined by eqs (Cll)

II.A [fig. 2].

App. D [fig. 12]. App. D [fig. 12]. IV.E [eq (40)].

II. A [eq (2)].

App. C [eq (C14)].

III. A [eq (21)].

[dimensionless] II. C. 2.

App. C [eq (C14)].

App. D [eq (D3)].

App. C [eq (CIO)].

vui
List of Symbols (continued)

Defined or used

Symbol

d

TT

p' (obs.)

fj',d',(t>'

Term

polar angle;

I. colatitude

McCamy's

notation

half-vertex angle vacuum

wavelength

obliquity factor

frequency

pi

reflectance

radial coordinate

BRDF

"tilted" coordinates

"rotated"

coordinates

(obs.)

Explanation

spherical direction coordinate; axes usually orient- ed so 6 is the angle from

the Z-axis, which is nor- mal to a surface element

dA

for a right-circular cone;

axis direction is ^o»<Ao> half-vertex angle is k

of right-circular cone (McCamy's notation) = c/f = cosd; dfJ. = — sin^-d^;

widely used, but not in this monograph

spectral or "carrier" fre- quency; I' = c/Kf) »

f

ratio of circumference to diameter of a circle ( =

3.141 592 65; solid angle

of l[hemisphere] =

277 [sr]; projected solid angle of l[hemisphere] = IT [sr])

distance from origin, in three-dimensional spherical coordinates fj,6,<t).

mistakenly called "bidirectional reflectance" or

"partial reflectance"

in [10]

[used in App. D only]

[used in App. D only]

in sec, eg, Unit-Dimension table, fig., etc.

[rad] II.A [fig. 2].

[rad]

[rad]

III.A [fig. 4].

III.A [fig. 4].

[m],[nm]or[/Am] I (footnote),

[dimensionless] II.B.l (footnote).

[Hz] or [THz] I (footnote),

[dimensionless] II.C.l [eq (19)].

[dimensionless] II.B.l [eq (15)]. [m] App. D only.

[sr-'] App. A [table 4].

[m], [rad], [rad] App. D.

[m], [rad], [rad] App. D.

mistakenly termed "direc- [dimensionless] App. A [table 4]. tional reflectance" in

[10]; should be directional-hemispherical reflec- tance, p(6i,(/)i;27r)

IX
List of Symbols (continued)

Defined or used

in sec. eg, Symbol

Pdr{Or,<t>r)

4)

SI

a

(0

Term

(obs.)

Explanation

radiant flux

azimuth angle;

longitude

projected

solid angle

solid angle

projected

solid angle

solid angle

mistakenly termed "direc

tional reflectance" in

[10]; should be hemispherical-directional re- flectance factor

Ri27T;er,(}>r) OT, by [12],

hemispherical radiance

factor /3 (277; 0r,<Ar)

power propagated as opti- [W]

cal electromagnetic ra- diation

spherical direction coordi- [rad] nate; axes usually orient- ed so that 4> is the angle

from the ^-axis in the

X—Y plane, tangent to reference surface at com- mon surface element dA = j cos 6- doj = jj cos 6- [sr] sin6-dd -dcf), as distin- guished from solid angle

u> elsewhere, often used inter- [sr] changeably with co used in [10], but we now [sr] recommend <^ = J/ sin 6

•dd-d(t> andn=J cos 6d(o = jj sine -dd -del) [sr]

Unit-Dimension table, fig., etc.

[dimensionless] App. A [table 4].

II.A [eq (1)].

II.A [fig. 2].

II.B.l [eq (10)]; II.C.l [eq (19)].

App. A [table 4].

App. A [table 4].

II.B.l [eq(lO)].

X
List of Subscripts

Defined or used

in sec, eg, Subscript Significance table, etc. d perfectly diffuse (isotropic; lambertian) App. C [eq (CI)].

/ intensity; e.g., the BRIDF/,/ III.C [eq (26)].

i incident II. A.

id ideal (lossless) and diffuse (isotropic or lambertian); e.g., /j,d App. C [eq (C7)]. = 1. isp ideal (lossless) and specular (mirror-like); e.g., p,s,, = 1. App. C [eq (C12)]. L radiance; e.g., the BRRDF/ri III.C [eq (25)]. m maximum; e.g., IV. C [eq (32)]. 0 direction coordinate of axis of right-circular cone in Mc- III. A [fig. 4]. Camy's notation

0 fixed coordinates (with respect to fixed axes) App. D [fig. 12]. r reflected II. A.

rl reflected-intensity III.C [eq (26)]. rL reflected-radiance III.C [eq (25)]. sp specular, regular (mirror-like) App. C [eq (C9)]. X denotes a derivative with respect to wavelength; e.g., Xk = IV.E [eq (40)]. dX/dk [X nm-']

List of Unit-Dimension Symbols

Symbol Name of unit Dimension or quantity Status in SI [38]

[g] gram mass base unit [Hz] hertz frequency derived unit with special name; = [s~' ]. [J] joule energy derived unit with special name; =

[m^ -kg s"^]

[m] meter length base unit [rad] radian plane angle {l[circle] = 27r[rad]} supplementary unit

[s] second time base unit

[sr] steradian solid angle {l[sphere] = 47? [sr]} supplementary unit [W] watt power, radiant flux derived unit with special name; =

[m^ -kg-s"^]

[X] unspecified (those of radiometric quantity X)

NOTE: the symbols are enclosed in square brackets to emphasize the dimensionality of the units and the usefulness of that dimensionality in routine unit-dimension-consistency checks and

analyses to cope with the great diversity of nomenclature in the hterature on optical radiation

measurements.

xi
List of Unit-Prefix Symbols [38]

SYMBOL PREFIX FACTOR SYMBOL PREFIX

a atto 1 A— 18 1,k kilo 1 A3 c centi 10"^ M mega 10" d deci 10 m "ii • mmi 1 A— 10 ' da deka 10' n nano 10-^

E exa 10'« P peta 10'5 f femto 10-'^ P pico io-'2

G giga 10» T tera 10'2 h hecto 10^ micro io-«

FACTOR NOTE: The symbols and prefix names listed in this table are used, in combination with the

symbols and names, respectively, of the SI units (see above) to form decimal multiples and submultiples of those units. For ex- ample, one terrahertz [THz] is equal to 10'^ hertz [Hz] or

10'2[s-']. Similarly, one kilo- gram [kg] is equal to 10^ grams

[g].

xii
Geometrical Considerations and Nomenclature for Reflectance

F. E. Nicodemus, J. C. Richmond, and J. J. Hsia

National Bureau of Standards,

Washington, D.C. 20234

and

I. W. Ginsberg

EG&G, Inc., Las Vegas, Nev. 89101

and

T. Limperis

Agro Sciences, Inc., Ann Arbor, Mich. 48103

A unified approach to the specification of reflectance, in terms of both incident- and reflected- beam geometry, is presented. Nomenclature to facilitate this approach is proposed. Under specified conditions—including uniform irradiance, a uniform, isotropic, plane surface, and

allowance for edge effects due to sub-surface scattering—the geometrical reflecting properties of a

reflecting surface are readily characterized or specified in terms of the bidirectional reflectancedistribution function (BRDF). The BRDF is denoted symbolically as fr:

fr(Oi,<i>i;Or,<i>r) = <fLr(0„0,;6»„(^,;f:,)/rff;,(e„<i)i) [sr-']

where

d and (f) together indicate a direction, the subscript i indicates quantities associated with incident radiant flux, the subscript r indicates quantities associated with reflected radiant flux, £, is incident

irradiance, Lr is reflected radiance, and d indicates a differential quantity.

The BRDF is a derivative, a distribution function, relating the irradiance incident from one given

direction to its contribution to the reflected radiance in another direction. Nomenclature (concepts, terms, symbols, and units) for categorizing and specifying reflectance quantities for a variety of

different beam configurations (both incident and reflected beams) is described, and all are defined and

interrelated in terms of the BRDF. The conditions under which the formalism can be apphed, including

situations involving considerable sub-surface scattering, are carefully established. The entire treatment

is limited to the domain of classical geometrical-optics radiometry and does not take into account

interference and diffraction phenomena, such as are frequently encountered with highly coherent

radiant flux. The other radiation parameters such as wavelength, (temporal) modulation, and polarization and the effects of fluorescence (or phosphorescence) are discussed briefly. Key words: Bidirectional reflectance-distribution function; diffuse reflectance; directional reflectance; nomenclature of reflectance; reflectance; reflectance factor; reflectance geometry; reflectance nomenclature; specular reflectance; sub-surface scattering; reflection; optical reflection.

I. Introduction

This monograph presents a unified approach to the specification of reflectance in relation to the beam geometry of both the incident and the reflected flux in any reflectometer or in any

application of measured reflectance data. Nomenclature to facilitate this approach is proposed.

Traditionally, optical propagation has been treated as consisting of two distinct phenomena

regular (specular) propagation and diffuse propagation. However, while purely regular (specular) or purely diffuse propagation can be very closely approximated, neither is ever completely and

independently achieved in practice. Furthermore, when directional propagance [1]' (transmittance

' Figures in brackets indicate the literature references at the end of this Monograph.

1
and/or reflectance) curves are plotted for actual measurement results, they not only exhibit a continuous distribution, from the pure delta-function spikes of highly specular propagance through

gradually broader and flatter peaks to the smooth flat curves of isotropically diffuse propagance,

but also show a wide variety of distorted shapes that do not fall directly between the two

extremes. An example of this is presented in our cover illustration that is repeated later in figure

5. The usual practice in reflectometry has been to express the reflectance properties of a real surface as the sum of a specular component and a diffuse component (with the possibly misleading

implication of being isotropically diffuse), and this is mathematically feasible. However, we feel that to do this is to make an unnecessarily artificial distinction, since the choice of what is included as specular and what as diffuse turns out to depend in many situations on the interests and objectives of the investigator or user and on the resolution capability of his instrumentation.

Accordingly, we propose to achieve greater generality and flexibility through the use of a single

bidirectional scattering-surface reflectance-distribution function (BSSRDF) to specify the geometrical reflectance properties of any surface. This approach is developed in this monograph where it is also the basis for a proposed nomenclature for more adequately describing and specifying the

(geometrical) reflecting properties of most surfaces in terms of the simpler bidirectional reflectancedistribution function (BRDF).

Note that the use of this nomenclature and approach does not preclude the use of quantities

associated with the concepts of specular and diffuse propagation, when appropriate. Whenever

attention is focused on a single ray (or the associated element of throughput) that retains its identity, for any purposes, along a given propagation path [2], its interactions with matter along

the path are usually most usefully described in terms of the propagance [1] (transmittance and/or

reflectance) or the attenuance [3] (absorptance and/or scatterance [3], the fractional scattering or

reflection loss) of the optical ray path. For example, this might be the path along each ray through an image-forming optical system between a pair of conjugate points on an object and its image.

Values of the BRDF at a point of reflection along a ray path in such a system with reflecting

optical elements can then be related to the "regular" or "specular" reflectance and/or the

reflection or scattering loss by "diffuse" reflectance in terms of the angular resolution capability of

the system. Values for directions within that angular resolution contribute to the "specular"

reflectance; those outside the angular resolution contribute to the loss or attenuation by "diffuse"

reflection or scattering. Of course the BRDF must be measured with angular resolution superior to that to which the data are applied. But that is a measurement problem that is not limited to this particular situation but applies to the measurement of many distributed quantities; and it does not

detract from the usefulness of the underlying concept, even in situations where only coarser measurement data are available. Reflection is the process by which electromagnetic flux^ (power), incident on a stationary^

surface or medium, leaves that surface or medium from the incident side without change in frequency;'' reflectance is the fraction of the incident flux that is reflected. There exists a large

volume of data on the reflectance, for optical electromagnetic flux, of a wide variety of reflecting surfaces. Most of these data are for the visible region, where the primary concern has been with

the visual appearance of reflecting surfaces, their color, texture, gloss, etc. More recently, a

substantial body of reflectance data for broader spectral regions, extending particularly into the

infrared, has also been collected, much of it in connection with heat-transfer analyses. And, very

recently, an interest in the possibilities for spectral analysis by reflected flux has arisen,

principally in connection with remote sensing of earth and other planetary and satellite surfaces. However, attempts to apply these data to particular situations frequently lead to difficulties and

inconsistencies. One very large source of difficulty is the lack of attention to geometrical

^ Radiant flux, luminous flux, photon flux, etc., can be substituted in the expression to be derived. Nomenclature (terminology, symbols, and units) used

here, unless specially defined or selif-explanatory, is that of ANSI Z7. 1-1967 [4].

' There would be a frequency shift in the flux from a moving reflector due to the Doppler effect.

' Radiation frequency v = cAo [Hz], where c [m s '] is the vacuum velocity of light and Ao [m] is the vacuum wavelength, as well as the fluctuation or modulation frequency f « v [Hz].

2
parameters, both for beams of incident flux and for beams in which reflected flux is collected and

detected. In fact, most optics texts fail to provide adequate nomenclature (terms and/or symbols)

for clearly designating and discussing these geometrical considerations. A unified approach to geometrical-reflectance and related nomenclature is proposed in this monograph.

Transmission, absorption, spectral and polarization effects, and fluorescence, essential

considerations for any complete treatment of reflectance [5], are not treated here. It should also be emphasized that we are concerned only with geometrical (ray) optics; it is assumed that there is no significant interference or diffraction. With the explosive increase in the use of coherent laser

flux, situations involving these phenomena of physical (wave) optics are becoming increasingly

important. Accordingly, it should be emphasized that the quantities and relationships developed in this paper should not be blindly applied to situations involving lasers. They may only be used

when it has been verified that, in fact, the basic assumptions of geometrical optics do apply to an adequate degree of approximation. Those assumptions are discussed in [6] and [7]. The remainder of this monograph is organized into four sections, in order to present first the

principal approach and then some variations on that approach and some related details. The main

ideas proposed by the authors are presented in section II. The bidirectional reflectance-distribution

function (BRDF) and reflectance concepts as well as the basic set of reflectance nomenclature are

formalized. Section III deals with some possibilities for additional reflectance quantities and

nomenclature. Practical considerations for measurement of reflectance quantities and nomenclature are discussed in section IV. Section V concludes with a summary and recommendations. In

addition, there are four appendices. Appendix A is a brief historical review of how we came to write this monograph and what transpired while it took so long to complete and publish it. It also gives the reasons for some of our choices of nomenclature. Appendix B describes a thought

experiment that some may find helpful in trying to understand the concept of the BRDF. The

expressions for perfectly diffuse and perfectly specular (regular) reflection are derived and

discussed in appendix C and the latter is also extended to off-specular peaks or "glints." The last appendix, appendix D, presents the detailed derivations of some important relations in McCamy's

notation.

II. Reflectance Concepts and Nomenclature

In order to simplify the development of concepts and formalization of nomenclature, we first study a model in which a relatively large area of reflecting surface is irradiated by a well- coUimated beam of incident radiation, examining the radiation reflected from a point well within

the irradiated area. Here, we confine our attention to geometrical considerations, to the effects of

just the spatial parameters of position and direction of all rays. Important effects of the other

radiation parameters—the spectral parameter (wavelength, frequency v, or wave number), the

temporal parameter (time, or fluctuation or modulation frequency / << v), and the polarization

parameter(s)—and of fluorescence (or phosphorescence) are briefly discussed in section IV. E.

A. Bidirectional Reflectance-Distribution Function (BRDF)

Consider the radiation flux incident on a surface from the direction {6i,<f>i), within the element

of solid angle rfw, [sr], as shown in figure 1. At denotes the total irradiated surface area. That

portion of the incident flux which strikes an element of area dAi [m^] centered at the point (xuyi)

will be denoted by d't>i [W]. The reflected radiance in the direction {Br,<i>r) at the point (Xr.Xr) which, comes from d^i will be called dLr. In general, dLr is directly proportional to </<!>,•, or dLr=Sd^i [Wm-2sr-»]. (1)

3
Figure 1. Geometry of incident and reflected beams (for general cases where sub-surface scattering is involved).

The factor of proportionality S will, in general, depend upon the location of the point at which the

incident flux strikes, upon the location of the point at which the ray of radiance dLr emerges, and

upon the directions involved, i.e.,

S = S{di,(t)i,Xi,yi;dr,(t)r,Xr,yr) [m-^-sr"']. (2)

This basic proportionality function S is called the bidirectional scattering-surface reflectancedistribution function, or BSSRDF, and it is a property of the reflecting surface. It is a particular

example of the more general scattering function S of [8].^ The treatment, so far, is completely general, with no simplifying assumptions other than those

of geometrical (ray) optics and the ignoring of other than spatial parameters. The function S

merely provides a way of quantitatively expressing or designating the connection between reflected

flux leaving (Xr,yr) in a given direction and the flux incident at (jCj.y,) from another given direction

(including, in the case of retroreflection, the same direction) which produces it. No assumption is made about the mechanism involved other than that there is some form of interaction between

radiation and matter by which some of the flux incident at {Xi,yi) emerges as reflected flux from

ixr,yr)- However, to obtain more tractable expressions, we will make some simplifying assumptions

that will still cover a very wide group of the cases of interest, at least to a useful degree of

approximation.

^ For more details about this important function and its significance, see below, section IV. C.

4
A convenient flat surface is chosen as a reference plane to represent the reflecting surface,

instead of attempting to describe and deal with its surface contours in microscopic detail. Then

the polar angles 6 are angles from the reference-plane normal, and azimuth angles </> are an^es

from an arbitrary reference direction in the reference plane. It is also assumed that the surface is uniformly irradiated over the entire part of the area Ai from which there is a significant

contributioin to the reflected radiance at {xr,yr), i.e., the incident radiance depends only on direction:

U=Ueu4>i) [Wm-^sr-']. (3)

The incident flux d^i on the element of area dAi [from just the solid-angle element in the

direction (0i,<^i)] 's

d^i = Li • cos di -doii -dAi

= dEi dAi [W], (4)

where dEi [= Li- cos di-doji'] is the incident irradiance and rfa>, is the solid-angle element within

which the incident radiance is confined. We can add up the contributions to the reflected radiance

at (xr,yr) from the entire incident flux in the direction {Oi,<i>i) and within the solid-angle element dojj

by integrating dLr [eq (1)] over the entire irradiated area:

dLr{0i,<i)i;0r,4>r,Xr,yr) = \ dLriOi,(l)i,Xi,yi;Or,<t>r,Xr,yr)

=

J S d<P, = dEi j S dAi = dEi \ S{0i,<t>i,Xi,yi;er,<l>r,Xr,yr)-dAi [W m-^ sr-']. (5) Ja,

If we further assume that the scattering properties of the sample are uniform and isotropic across the reference plane, the scattering function S does not depend on the location of the point

of observation (Xr,yr), but it still depends on the distance r between (Xi,yi) and {Xr,yr)- Equation (5) can now be written as

dLr = dEi •friei,Mr,(i>r) [W m-^ Sr-'] (6)

where

fr= \ S{di,<i>i;dr,ii>r;r) dAi [sr"'] (7)

and

r = [{Xr-Xif -^{Vr-yin^. (8)

Thus, for uniform irradiance over a large enough area of a uniform and isotropic surface, the

basic quantity that characterizes (geometrically) the reflecting properties of that surface is the

function fr:

Me,,<t>i;dr,4>r)^dLridt,<l>i;er,<l>r;E,)/dEi{ei,<l>i)

^dLr{ei,<f>i;0rAr;Ei)/[Li{ei,<l>i)-COS0i-d(Oi] [sT'']. (9)

5
The name that we propose for this function fr is the bidirectional reflectance-distribution

function, abbreviated as BRDF [9]. Since for a given pair of directions, the BRDF fr is a concentration of reflectance (per steradian) it may take on any value from zero to inflnity. Also,

since fr depends only upon directions, we can represent the geometry of the incident and exiteut

radiation by a simple polar diagram as in figure 2. Nomenclature for categorizing and specifying reflectance quantities for a variety of different beam configurations is defined and eill such quantities are interrelated in terms of the BRDF in the

following sections.

The use of infinitesimals in radiometric analysis, as in the above definition of the BRDF, is discussed and explained in detail in [2] and by Jones [7].® The BRDF itself, as a ratio of

infinitesimals, is a derivative with "instantaneous" values that can never be measured directly.

" See, also, section IV. A.

z

X

Figure 2. Geometry of incident and reflected elementary beams,

(Z-Axis is chosen along the normal to the surface element dA at 0.) (adapted from [10]).

6
Real measurements involve non-zero intervals of the parameters, e.g., Aw or A\ rather than du) or dk, and, hence, can yield only average values fr over those parameter intervals. But this is true of many basic physical quantities. A speedometer never truly shows instantaneous speed, only an

average over a period of the order of its time constant, but that does not invalidate the usefulness

of the underlying concept of instantaneous speed.

B. Definitions of Reflectance and Reflectance Factor

1. Reflectance

Reflectance is the ratio of reflected to incident flux. From conservation of energy, it follows

that reflectance may have values only in the interval 0 to 1, inclusive. As, previously mentioned, a

derivative quantity, such as the BRDF fr defined in eq (9), is useful primarily as an underlying

concept, but it can never be measured directly because truly infinitesimal elements of solid angle

do not include measurable amounts of radiant flux. Actual measurements of reflectance quantities

always involve non-zero intervals of the governing parameters (in this case, the spatial parameters

of position and direction). We need, then, a general expression for reflectance for cases where the geometry of the incident and reflected beams can be specified arbitrarily, and this can be written

readily in terms of the BRDFfr(di,(t>i',(^r,4>r)- In order to economize on writing we will represent the product of an element of solid angle

doj with the cosine of the angle 9 between the normal to the surface and the direction associated

with dco by dO. (dCl = cosd dco). The quantity dCl is commonly called an element of projected

solid angle. {See appendix 2 in [2].}'^ In general, the radiant flux incident through a solid angle o), onto a surface element dA is

d<^i=dAl L,(e,,</),)-rfn, [W]. (10)

J Oil

Similarly, the flux reflected by the surface element dA into a solid angle Wr is

d(^r=dA l Lr{6r,(t>r)-dnr [W], (11)

which can be written as

d^r^dA-l

I Md,,(t>r,er,(l>r)-Li{d^,4>i)-dClrdnr [W] (12)

since

LAer,<J>r) = I dLAe^,4>i;dr,(f>r;Ed (13)

and, from expression (9),

dLr{eu<t>i\er,4>r;Ed = Wu<i> i^S^ r) ' UdM' d^i . (14)

A general expression for the reflectance of a surface with an arbitrary configuration of beam

geometry with regards to direction is obtained from the ratio of eq (12) to eq (10):

' Another approach is to use the notation = cosH , dfi = -sinH . Then du) s -sinH dU d<t> = -d/i dii . elc. [W]: but it is not adopted here.

7
f)io)r,(Or;Li) =d<t>r/d<Pi (15)

LiidM-dfli

[dimensionless]. (15a)

Up to this point, we have only placed the condition upon L, that it be uniform over a

sufficiently large area, and have not placed any conditions on its variation with direction. If we

now specify that the incident radiation be uniform and isotropic within the incident beam, the constant value of L, can be brought outside the integrals in both numerator and denominator so

that it cancels, leaving the biconical reflectance

.

p(a;,;a;,) = (1/a,.)-

j | /,(ei,0i;0„0,)-dfii-rfn,

[dimensionless], (16)

where

ft,- =

I dili

.

The assumed condition, that L, is the same at all points and in all directions within the incident

beam, is fairly well approximated in any well-designed reflectometer, so this expression, given in eq (16), is very useful for describing the observed reflectance of a surface under various

configurations of beam geometry. Note, however, that it is important, always, to specify the

geometry of both the incident and reflected beams associated with a given value of reflectance.

2. Reflectance Factor

A reflectance factor is defined as the ratio of the radiant flux actually reflected by a sample

surface to that which would be reflected into the same reflected-beam geometry by an ideal

(lossless) perfectly diffuse (lambertian) standard surface irradiated in exactly the same way as the

sample. In eq (12), we already have a general expression for the reflected flux d^r of any sample

surface element, characterized by its BRDF fr(0i,(t>i;0r,4>r)- This same equation will also give the

reflected flux d*t>r,ici of an element of the ideal standard surface for fr,id = W [eq (C8), app. C].

Then the ratio is

dA-l

J fABu<i>uer,<l>r)-ueiM-dVLrd^r

d<t>r/d^rM = — — • - (17)

{dA/n)-\

I Li(di,(t>i)-dnrdn,

If, as before, we make the assumption that L, is constant (isotropic within the full solid angle of

incidence oj;), L, can again be brought out from the integral sign in both numerator and

denominator where it cancels along with dA, leaving as the general expression for a biconical

reflectance factor

/?(w,.;a>,) = [7r/(n,.n,)]- ( I fr(di,(t>r,dr,<f>r)-dn, dilr

OJ

J, (jj^ [dimensionless]. (18)

8
In some cases, the use of the reflectance factor is preferred over the use of reflectance. For

example, in the case of almost lambertian reflectors, the reflectance factor R is nearly independent

of beam orientations, thereby making the reflectance factor a particularly attractive quantity to use

in describing such reflectors.

C. Reflectances and Reflectance Factors for Nine Geometries

1. Reflectances for Nine Geometries

The proposed nomenclature for nine kinds of reflectance is presented in table 1. Listed there are the proposed term, the proposed symbol, and the definition in terms of the bidirectional

reflectance-distribution function (BRDF) /r(^,,</),;^r,</>r) for each kind of reflectance, following, with some modifications, a scheme presented in [12]. The geometry for the designations used in the

terms of table 1 is illustrated in figure 3. Directional denotes an element of solid angle dw about a

single direction {6,4)); conical denotes a solid angle co of any configuration (the common special case where w is a right circular cone will be discussed in section III. A); and hemispherical

denotes a full hemispherical solid angle cu = 277 [sr], for which the corresponding projected solid angle

.- .-277 .-7J-/2 n= cosO-doj^l COS d- sin d dd d(t> ^-n- [sr]. (19)

J 2TT Jo Jo

If we permit the solid angle a» to include the extreme values, so that dci) < w < 277 [sr], then eq

(16) (relation 5 in table 1) contains all of table 1. The corresponding expressions for all nine

reflectances listed in table 1 are formed merely by substituting the appropriate values for w, and

o)r in eq (16). Note that each reflectance is formed simply by integrating the BRDF fr(di,4>i',6r,(f>r) with respect to the projected solid angle Or over the appropriate a»r and averaging it with respect

to the projected solid angle fi, over the appropriate a»,.

It should also be emphasized that the foregoing reflectances are applicable only to situations

with uniform and isotropic radiation throughout the incident beam of radiation; i.e., where the

incident radiance L,(0i ,(/>,) = L, is a constant, with the same value at all locations and for all directions (di,4>i} included in the incident solid angle a>,. If this is not true, then one must refer to the more general expressions, such as eq (15) for non-isotropic uniform incident radiation. In the same way, note that these quantities are given only for a uniform reflecting surface. They are applicable to extended areas of non-uniform surfaces when it is adequate to obtain just the

average properties over such areas. In this case, it is important that there be uniform irradiation of

all parts of the surface, from which is produced the flux reflected from the designated area. Otherwise, the result will be a weighted average, according to the distribution of incident

irradiance as a function of position, for each direction involved. The considerations discussed in these last two paragraphs with respect to reflectances are equally applicable to reflectance factors,

multivariate reflectance-distribution functions (MRDPs), and other reflectance quantities described

and discussed throughout this monograph.

2. Reflectance Factors for Nine Geometries

The proposed nomenclature for nine kinds of reflectance factor is presented in table 2, which

follows the same format as table 1. Here we have departed from ANSI Z7. 1-1967 [4], and from the CIE-IEC International

Lighting Vocabulary [14] on which most of [4] is based, in recommending that the term

"reflectance factor" and symbol R be used for all nine of these quantities, including three for which the term "radiance factor" (or "luminance factor") and symbol /3 are now the accepted

standards. We make this recommendation because we feel that the proposed scheme is adequately

Table 1. Proposed nomenclature for nine kinds of reflectance*

1. Bidirectional reflectance df)i0i,4>i;dr,'i>r) = fri.Oi><t>i\Or,<}>r) ' di\

2. Directional-conical fj(d,,(t>i;cjjr) = I fr(Oi,(t>i'firJ^r) ' d(ir

reflectance^

3. Directional-hemispherical p(Si,<^)j;27r) = j fr[0i,(l>i;0r,<t>r) ' dClr

reflectance ^"

4. Conical-directional dfj{oji:0r,4>r) = {dnr/n,) \ fr{0i,<l)i-0r,<br) dCl, reflectance^ ""i

5. Biconical reflectance^ f}{w,:Wr) =(l/ni) j j)

• dfi^ -rfdi

6. Conical-hemispherical P((^i:27T) =(i/n,)-| j

fr{0t,4>r,Or,<t>r) dilr dil^ reflectance^

7. Hemispherical-directional dp{2n;er,<t>r) =(di1r/n)- j f(ei,<i)i\0r,<i>r)-dVli reflectance

8. Hemispherical-conical p(27T;ajr) =(\Itt)- I I fr{8j,4>i;9r,(t>r) ' d^lr dilj reflectance^

9. Bihemispherical p(2tt:2tt) =(1/77)- [

( f(Oi,4>i;0r,<t>r)-d(lr-dn,

reflectance -^z"

adapted [9] from [12].

^ When, as happens most frequently, a "conical" solid angle co, lying between the extremes of an element d(o and a full hemisphere, is in the form of a right circular cone of half-vertex angle k, with its axis in the direction (^o-'^'o), 'his can be

specified more explicitly with a notation suggested by McCamy [13] (6'o,<^o.'<)- For example, if this is true of both the sohd

angles of incidence and reflection, the biconical reflectance f){(Oi:(Or) could be written more explicitly as (see section HI. A)

P(Soi,<^>OM'<i;^Or,<^>Or,'<r)

Note: The symbol oj is used here to designate solid angle, and fi to designate projected solid angle:

(jj = jdoj =

j lsin0-de-d(t>; il = j dn = j cos9-dw = \ j cosd-sind-dd-d(t>

clear and explicit and because it is more consistent with the close interrelationship between all nine quantities. On the other hand, we also favor retention of the present terms as acceptable

alternates to the proposed basic scheme. There are times when it is convenient to be able to refer

to items 1, 4, and 7, of table 2, collectively, as the "radiance factors" R{oji\9r,(t>r) or /?(aj,;dajr), an

expression that becomes quite awkward as the ". . .-directional reflectance factors." Again, as in table 1, the biconical quantity [eq (18)] can be considered as the basic one. The

(biconical) reflectance factor is just tt [sr] times the average value of the BRDF fr, averaged over

the designated solid angles of both incidence and reflectance (collection) with respect to projected

solid angle. (The reflectance factor is a pure dimensionless ratio while the BRDF has the

dimension [sr"^], which is cancelled when its average value is divided by 1/v [sr"*], the BRDF of

a perfect lambertian reflector {equivalent to multiplying by tt [sr]}.) And again, all eight of the

remaining reflectance factors can be formed by substituting i6,<j)) or du), as appropriate, for a

directional quantity, or the other extreme value 2tt [sr] solid angle for a hemispherical quantity,

for either one or both of the solid angles and cOr in eq (18).

11
Table 2. Proposed nomenclature for nine reflectance factors*

1. Bidirectional reflectance R{dt,<t)i;dr,4>r) = '"'fr(0i.<t>i;6r,(l>r) factor*"

2. Directional- conical J? (e,>,;a),) = (n/nr) \ fr{0i,4>i;6r,4>r) dD.^

reflectance factor^ ^'"r

3. Directional-hemispherical (0, ,</)j ;27r) = I fr(ei,4>i;0r,4>r) dClr

reflectance factor f

4. Conical-directional R {c^r,0r ,(l>r) = (n/Cli) I fr{d-:,(t>r.Or,({>r) dClf reflectance factor^ *"

Biconical reflectance R((Ot;iOr) = /(H, |

I /^(e, ,(/),; </>r) • rfH, (fn, factor^ L L_

6. Conical-hemispherical /?(w,;27r) =(1/(1,) / / fr(di,<f)i;lir,<t>r) dVl, di\

reflectance factor^ '2''

7. Hemispherical-directional R {2n:0^,<l>,) =| fr{Oi,(j)rMr,<l)r)- dfli reflectance factor*"

8. Hemispherical-conical R{2TT-Wr) =(l/n,) j

I fr(ei,<t>r,Hr,4>r) dflr-dClf

reflectance factor^ iu,

9. Bihemispherical «(277; 277) =(1/77) / I fr(ti,,(t>r.Or,<t>r) di^ di^

reflectance factor -^f

adapted [9] from [12].

^ (See footnote "a" of table 1.)

"" Judd [12] calls these three quantities "radiance factors" (in accordance with [14]) and uses the symbol (3 (instead of R). The recommended change to "reflectance factor" and the symbol R for aU nine quantities is discussed briefly in the text. Note: (See note following table 1).

III. Some Additional Reflectance-Nomenclature Possibilities

We have presented the BSSRDF and BRDF, the basic quantities for characterizing and

specifying the (geometrical) reflecting properties of a surface, and have related them to the widely

used reflectances and reflectance factors (including radiance or luminance factors). Now we want

to present some other modifications and extensions to this scheme of nomenclature that, although

they may not be so essential or of such wide application, nevertheless may have substantial

usefulness for filhng more limited needs.

A. McCamy's Notation for Right Circular Cones

As illustrated in figure 3, the solid angle oj denoted by the term conical, as used here, may

have any configuration and is not limited to a right circular cone. On the other hand, with

cylindrically symmetric optical systems, the commonest form of beam for either oj, or (Or is a right

circular cone. Accordingly, for this common special case, McCaray has proposed [13] that such a cone be specified by its half-vertex angle k and the angles, in spherical coordinates, for the

direction of its axis (through the origin) do,(t>o (see fig. 4). Thus, a biconical reflectance where both

beams are right circular cones would become (see appendix D)

Pi^Oi,(t>Oi,Ki;dor,(t>Or,Kr) = (tT • sin^K; COS^Oi)"'

.

£/</)r "cos^r 'sin^r 'ddr rf*/), • cos^j -sin^, 'ddi. (20)
where

a{6) = cos~'[(cosK — cos0o'cos^)/(sin0o'sin^)]- (2

A similar expression can be written for the biconical reflectance factor, or for any conii reflectance quantity, where the beam is a right circular cone.

13
COS^r sin^r ' <^<Ar " dQr ' COsdi • sin^i • (i</)j • d^i, (22)

where, as before, a{6) is given by eq (21).

B. Multivariate Reflectance-Distribution Functions (MRDF's)

Two of us, (Ginsberg and Limperis), feeling a need for more explicit subdivision of the broad

biconical category, proposed the multivariate scheme of reflectance-distribution functions shown in table 3. Reflectances and reflectance factors can also be categorized in the same manner. However, the most useful quantity for extrapolation to different beam geometries is the average

reflectance-distribution function, so it is that quantity that is tabulated here. In table 3, in addition

to listing the proposed term, symbol, and defining relation (in terms of the BRDF or 4RDF), there

are also small pictorial diagrams to aid in visualizing the beam geometry for each case. The multivariate reflectance-distribution functions (MRDPs) are grouped according to the

number of independent variables as: (1) quadrivariate—all four independent variables (^,,</),;^r»<^r) govern the bidirectional reflectance-distribution function BRDF or quadrivariate reflectancedistribution function 4RDF (these designations are completely synonymous); (2) trivariate—any

three of those four variables are still independent variables for the trivariate reflectance-distribution

functions 3RDF's that have been averaged, with respect to the corresponding component of the

projected solid angle, over the full range of the remaining variable; (3) bivariate—any two of the

four variables are independent for the bivariate reflectance-distribution functions 2RDF's that have

been averaged, with respect to the corresponding components of the projected solid angle, over

the full range of each of the remaining two variables; (4) univariate—any one of the four variables

is still an independent variable for the univariate reflectance-distribution functions IRDF's that have been averaged, with respect to the corresponding components of the projected solid angle, over the full range of each of the remaining three variables; (5) avariate—the avariate reflectancedistribution function ORDF is single valued, with no independent variables, having been averaged

over the full range of each of all four variables (the bihemispherical case), with respect to projected solid angle.

There is an explanatory note at the end of the table concerning the functional notation for designating directionality for the MRDF's. The situations where an independent variable is assigned its maximum value {6—77/2 [rad] or (f>=277 [rad]) are explicitly distinguished from those

where the function is averaged over the full range of that variable {6 from 0 to 77/2 [rad] or 4> from

0 to 27T [rad]) as well as from those where _the function is averaged over the full solid angle of a hemisphere (en,, =277 [sr]). For example, /r(77/2,27r;7r/2,277) denotes a particular value of the

bivariate reflectance-distribution function 2RDF /r(0,,277;7r/2,(/)r) that has been averaged (with

respect to projected solid angle) over the full ranges 0 to 27r [rad] for c/), and 0 to 77/2 [rad] for Or

[(3.4) table 3], evaluated at 6^—77/2 [rad] and (/)r = 277 [rad]. Another example is the conicalhemispherical reflectance-distribution function CHRDF /r(ct»,;277), which denotes a reflectancedistribution function where the BRDF has been averaged (with respect to projected solid angle) over an unspecified (conical) configuration of incident beam and over a reflected (collected) beam

filling a full hemisphere of ojr = 277 [sr]. Each MRDF is an average of the quadrivariate

(bidirectional) reflectance-distribution function (4RDF or BRDF), with respect to the corresponding

component of projected solid angle, over the full range of each of the angles not shown as independent variables. For example, the bivariate reflectance-distribution function (2RDF)

fr{Qi,27T ;77 12,4>r) is the average of the 4RDF or BRDF taken over the fuU range of each of the

remaining angles 4>i (0 to 27r) and dr (0 to 77/2) with respect to cosdr sindr ddr d(f)i. Thus:

14

CO

16
•o u

T3

17
s1) (0

03 t: 4) t3

Ml'- § 'as

Ql C«

9- «

.S Si^ t3

^ I

i2 u u CO

G

56

^ Ml

I I* " Eg

4^ c at) Qi, _

(0 « ec; ^

SI {fl CO

1=^ u

-c J2 .!£

Q

DSo

•c

IN

.2 '2 O T3 3 ,

S 1

J3

I<1 J2

C O 5 a =i ^ J2

CO S o t-

.S -o

Js a.

be V

.5

eft B

4^ C

CO c(9 o

§

o

4) CO P

« "3

-bbB V O " B 4J

CD .B O

'C o

18
r2n rnl2 fri0i,<l>i;Or,<}>r)-COsdr-sindr-ddr-d<t>i _ •'O •'0 ~ r27r r7r/2

I I cosdr- sindr'ddr'd<{>i

'0 -'0

[sr-']. (23)

But

2n r 77/2

I d4> = 2tt [rad] and I co&d -smd -dd — Va [rad], so

-'0 -'o

J

"217 ririz friei,(}>i;dr,(f>r)^COsdr-Smdr-ddr-d4>i [sr^. (24)

0 •'o

As was previously mentioned, the reflectance factor for uniform isotropic irradiation is equal to the

BRDF averaged with respect to the projected solid angle over the specific sohd angles of incidence

and reflectance involved and divided by (l/7r)[sr~']. Since the MRDF's are similar averages over

specific sohd angles, one may simply multiply an MRDF by tt [sr] to obtain the value of the

reflectance factor for the corresponding geometry.^

C. Other Special Types (BRIDF and BRRDF)

The McCamy notation (section 111. A) is specific to the choice of right-circular-cone solid angles, and the MRDF's (section III.B), as given, are specific to the polar coordinate system by

which they are defined. Other specific solid angles are also sometimes used. As an example,

consider the measurement of gloss, in which the geometry involves specified rectangular-conical

solid angles in a special case of biconical reflectance.

In glossimetry, the measurement of luminous specularity, the value of fr varies significantly

with angles of reflection making it a very sensitive measure of gloss. Specifications have been

established for the measurement configuration [15, 16, 17] including the orientation and magnitude

of the rectangular-conical solid angles of irradiation and reflection, and the characteristics of the

light source and the detector. Since gloss is defined in terms of rectangular-conical solid angles, it may be more convenient to express the BRDF in terms of a coordinate system in which the

integrals over these solid angles can be more easily evaluated. However, the meaning and

usefulness of the concepts of BRDF and projected solid angle are retained even when they are expressed in other coordinate systems.

Occasionally, situations arise where a reflecting surface is viewed from a great distance where

it appears to be, for all practical purposes, a point source. This occurs, for example, in imaging

systems when an object cannot be resolved because it subtends an angle less than the resolution

limit of the system. In such circumstances, it is not the radiance L, but rather the radiant

intensity / = j L -cosd • dA, of the object that is of significance, and, in deeding with reflected radiation from the object, the significant distribution function is one giving the directional

distribution of reflected intensity.

This has been ignored up to this point and emphasis has been placed on the directional

distribution of reflected radiance, because the reflected-intensity distribution is too often used when we beheve the reflected-radiance distribution is more appropriate. The latter is the more basic,

^ See last paragraph of section 11. B. 2.

19
because, when the radiance is known, the intensity can always be obtained from it as / = j^L • cosd dA [W sr"'], while the reverse is not true. Given only the intensity /, this integral equation cannot be solved for the radiance L without more information about its distribution over the

radiating (emitting and/or reflecting) surface in question.

When it is necessary to distinguish between these two reflectance-distribution functions

(RDPs), the following terminology and notation may be used. The bidirectional reflected-radiancedistribution function (BRRDF), which we have been calling just the BRDF up to this point,^

frL{Qu^uer,<i>r)^ dLr{ei,(t>i-,dr,4>r-^i)ldEi{eu(t>d [sr"'] (25)

while the bidirectional reflected-intensity-distribution function (BRIDF) is

fr,{eu4>uer,<^r)^dIr{eiA-A,(f>r\^i)ld^i{eu(i>i) [sr-1]. (26)

DeVos caUs this quantity the "partial reflectance" r''" [19]. For a surface element dA, small enough

so that there is no significant variation of radiance L across its surface, it is easily shown that

frl{ei,ct>i;dr,4>r) =frL(ei,<l^i;er,<t>r)-COsdr [sr""]. (27)

It may also be useful to express reflectances and reflectance factors in terms of the BRIDF/r/. As we have seen, it is adequate to have just the expression for the biconical quantity, since the other

eight follow readily from it, in each case. The biconical reflectance, then, is

pK,a;,) -

I fr,{di,c}>r,dr,(l>r)-dwr-dnr, (28)

and the corresponding biconical reflectance factor is

(to,., a;,) - [77/(0, -ft,)]-

j I fr,(di,(l>r,dr,<l>r)-d(Or-dni. (29)

The BRIDF also figures prominently in an ingenious approach to modelling the reflectance of any

rough surface in terms of an equivalent, single, optically smooth, curved surface of revolution, by

Trowbridge and Reitz [20]. In the balance of this monograph, and elsewhere, we will continue the practice of using the

simpler nomenclature bidirectional reflectance-distribution function (BRDF) f0i,(l)i;dn<t>r) in lieu of the longer bidirectional reflected-radiance -distribution function (BRRDF) unless otherwise

explicitly stated.

IV. Practical Considerations for ]\feasurement of Reflectance

So far, the proposed scheme of reflectance nomenclature is described primarily in relation to an idealized abstraction, reflection by a surface element. Concerning the application of this proposed reflectance nomenclature to real situations, we are going to point out severed areas where

caution is needed: effects of finite intervals of area, angle, solid angle, and distribution function;

definition of reflecting surface area; effect of sub-surface scattering; effects of other radiation

parameters such as wavelength and polarization and of fluorescence (or phosphorescence); and the

advantage of using reference standards of reflectance.

' The term "relative radiance" and symbol l/e,<t>;9' ,<t>') have also been used for this quantity [18].

20
A. Effects of Finite Intervals

There are some very general considerations, of wide application to many types of physical

measurements, that are involved here when we speak of a differential quantity, such as an

(infinitesimal) element of area, of angle, or of solid angle, and of the closely related concept of a

derivative, an instantaneous rate, or a distribution (function), such as irradiance or exitance

[W-m~^], radiance [W-m~^ -sr"'], or reflectance-distribution function [sr~']. It must always be

recognized that none of the latter can ever be measured exactly, just because any real measurement must be made over a finite interval (neither infinite nor infinitesimal) of each

independent variable or parameter governing the quantity or its distribution, so that the result,

necessarily, is an average viilue over each such interval. This is true even in situations where, at

first glance, it may seem that instantaneous rates are being measured quite directly. For example,

an automobile speedometer is actually fairly sluggish and, when speed is changing, responds

rather slowly to those changes, so that its indication is always the average value over some small

time interval, of the order of its time constant or response time. Similarly, for a doppler-effect

measurement of velocity, a certain number of cycles, possibly only a fraction of a cycle in highly

sophisticated instrumentation, must be received in order to establish a value for the frequency

shift in hertz, placing a limit on the speed of response. And there is always a resolution limit, related to a noise level or a frequency-response characteristic, that establishes an interval over which averaging or integration takes place, so that a truly "instantaneous" measurement, though a

highly useful concept, is never actually possible.

For slowly varying distribution functions, where it is possible to measure over parameter

intervals small enough so that significant variations do not occur within the interval, this is not a problem. However, when a distribution fluctuates more rapidly or, in the limit, becomes

discontinuous, problems arise. For example, we have seen that, in specular or near-specular

reflection, the bidirectional reflectance-distribution function (BRDF) rapidly grows very large,

approaching infinity in the ideal case, in the specular direction. This is illustrated in figure 5. Note

that a logarithmic scale was used in order to portray both the very high peak value and the

interesting structure in the very low values in other directions at the same time. Such

measurements of the BRDF with a gonioreflectometer may be so sensitive to the angular-resolution

capability of the instrument as to make the resulting values, averaged over that resolution interval,

of httle significance. This is analogous to trying to measure absolute spectral line profiles with a spectroradiometer of inadequate spectral resolving power. In such cases, while the concept of a BRDF that approximates a delta function is still useful for understanding and relating one situation

to another, values of reflectance or reflectance factor that are proportional to the integral or average of the BRDF over the projected solid angle of reflection, for appropriate sizes and

configurations of that solid angle, are probably more useful. Although we will not go into the details, it should be noted that the methods of Fourier

analysis [21], developed to a large degree in connection with electrical communications, and the

broader related developments of information and communication ' theory and statistical analysis, can be very helpful in understanding and dealing with interactions between distributions and

measuring instruments. By treating an incident distribution as an input signal, and expressing the

characteristics of a measuring instrument in the appropriate way as a response function

(mathematically relating the resulting output to the input signal), it is theoretically possible to reconstruct the input, from the output, to any desired degree of resolution, depending on the

extent to which the output and the system response function are known. However, in reality, there

is always some noise level that places a Hmit on the specification of these quantities and hence

Umits the achievable reconstruction of the input, causing an irretrievable loss of information. The

degree of this loss and the extent to which degradation of resolution by instrument characteristics

is recoverable can be evaluated.

21
^ - o' —• f

' 180*

to 10* 10"* 10"" f - 10"' 10"* lo' 10

10"' 10"' f =10-* 10** 10** 10"' f = 10"^ 10"*

Figure 5. Three-dimensional graph of bidirectional reflectance-distribution function (BRDF)fr {33.2", 0°;6,(l>) (from [10]).

22
Explanation of Figure 5

This represents a three-dimensional graph (surface) of the bidirectional reflectance -distribution function /r(33.2°, 0°; 6, (f)) of a sample of aluminum. It shows the directional distribution of

reflected radiance [W -cm"^ -sr"'] per unit incident irradiance [W cm~^] in a well-coUimated beam

from the direction 0, = 33.2° (angle from normal to reflecting surface) = 0° (azimuth angle in plane of surface). It was prepared by Nicodemus as a cover illustration for Applied Optics to go

with [10]. Data reported by Gerald M. Keating and James A. Mullins in "Vectorial Reflectance of

the Explorer IX Satellite Material," NASA Technical Note NASA TN D-2388 (Aug. 1964), in the

form of a contour plot of bidirectional reflected-intensity-distribution function (BRIDF; see sec. III.C)/r/(33.2°, 0°; e, 0) = /rt(33.2°, 0°; d, 0)-cos^, were transformed into polar plots of /r = fri

in vertical planes at different azimuth angles (p. Artist James Cutter at Sylvania Electronic

Defense Laboratories fabricated aluminum fms cut to the shapes of these polar graphs and

assembled them to produce the modernistic sculpture which was photographed. The graph is the

surface which forms the envelope of the curved sections. Examples of individual vertical-plane

sections are also shown. The logarithmic scale for fr exaggerates the variations in the low values of

fr in directions away from the strong specular reflection at (33.2°, 180°) to bring out the fact that any description as a combination of specular reflection (S-function) and perfecdy diffuse (lambertian)

reflection (constant in aU directions) cannot tell the whole story. In commenting on the peculiar

ridges observed in this pattern, Keating and Mullins state that "It was suggested . . . that this effect was probably due to the cross grain of the [aluminum] material."

'Figure 5 also appears on the cover.

23
In addition to the problems of achievable resolution in measuring rates or derivative

quantities, there is also the problem (still geometrical) of interaction between position and direction

parameters, known as vignetting. Ostensibly, we should be able to extend the relations stated for a surface element to a larger portion of a plane reflecting surface merely by arranging for uniform

isotropic irradiation and for reflection (collection or viewing) through the same solid angles or

directions, respectively, for all elements of the more extended surface. However, with practical

laboratory-size instrumentation, sources and receivers are not at infinity, although they can be

optically at infinity if they are each positioned in the focal surface of a coUimating or collecting

optical system. But this, in turn, introduces still other problems.

The imperfect reflectance and/or transmittance of the optical elements of a collimator, and its aberrations, which may produce significant departures from perfect coUimation or focusing, are not easy to control or to evaluate for their effects on a reflectance measurement. Similar

considerations apply to the receiver and its associated collecting optics, if any. Even with ideally

perfect optics, directional variations in the coUimated beam will be produced by any variations in radiance across the surface of a source located in the focal plane; and the size of the solid angle

filled by the collimated beam will depend on the size of the source (the solid angle that it subtends

at the primary optics). Similarly, uniformity of radiance across the collimated beam (across the exit pupil or aperture of the collimator) depends on the radiation being isotropic (lambertian) at each

point of the source. Since extended sources of uniform radiance that are isotropic over large solid angles are difficult to obtain, the problems involved in using a collimator become even greater for the larger solid angles of irradiance when "fast" optics (of low f number) are used. Similar

considerations apply to the receiver and its associated collecting optics for the larger solid angles

of collection (of reflected radiation).

It is particularly hard to achieve uniform, isotropic irradiation and/or collection of reflected

flux from a fuU hemisphere above an extended flat reflecting surface, even without considering the

unavoidable problem of spatial overlapping of the two beams when one (or both) of them fills the

full hemisphere. The integrating sphere probably offers the most satisfactory approach from the

standpoint of geometry alone, but it introduces problems of achieving adequate power levels for detection and even more serious difficulties due to spectral variations in the reflectance of

available wall materials. Again, we do not go into details but merely call attention to the possible

difficulties in extending the nomenclature and concepts to real situations involving extended

reflecting surfaces.

B. Definition of Reflecting-Surface Area

Another geometrical problem is that of physically defining the exact portion of reflecting

surface entering into a measurement. This can be done in three ways, deferring until the next

section the major problem of sub-surface scattering effects: (1) by a sharply focused irradiating

beam; (2) by a sharply focused collecting beam; or (3) by the extent of the (exposed) reflectingmaterial surface itseF for non-sub-surface scattering conditions (see fig. 6). In (1) both the

reflecting-material surface and the area from which reflected radiation may be collected by the

receiver must, of course, be greater than the area defined by the focused irradiating beam at the

largest obliquity (maximum value of 0,) that is used. Similarly, in (2) the area irradiated and the

extent of the sample-material surface must be clearly greater than the maximum area from which

the focused coUecting beam receives reflected radiation at the largest obliquity (maximum value of

dr). In both of these cases, sharp focusing also implies adequate stops and baffles to insure

against transmitting or receiving any rays outside of the defined beams. For the moment, we

ignore the added problems of aberrations and adequate depth of focus. Finally, in (3) both beams

extend beyond the limits of the (exposed) reflecting-material surface and extreme ingenuity must

be exercised to devise arrangements where the measurement is not falsified by additional stray

reflections from supports, housings, and other unwanted sources. Alas, the convenient cold black

non-emitting and non-reflecting material of our thought experiment, described in appendix B, is not available in practice.

24
i - WIDTH OF INCIDENT BEAM [DEFINES REFLECTING AREA IN (1) ].

r - WIDTH OF REFLECTED BEAM (THAT CAN BE ACCEPTED AND MEASURED) [DEFINES AREA IN (2) ].

s - WIDTH OF EXPOSED SAMPLE SURFACE [DEFINES REFLECTING AREA IN (3) ].

ONLY THE EXTREME RAYS OF EACH BEAM ARE SHOWN.

SOLID RAYS MUST BE SHARPLY FOCUSED TO DEFINE THE REFLECTING AREA. DASHED RAYS NEED NOT BE SO SHARPLY FOCUSED.

IN CONFIGURATION (3) IT IS IMMATERIAL WHETHER i>r OR Kr ORi^r.

Figure 6. Three configurations for physically defining the reflecting area.

(See section IV. C concerning important limitations arising from sub-surface scattering or "edge effects".)

Unfortunately, there are unavoidable conflicts between some of the foregoing considerations

concerning the coUimating and collecting optics. In order to avoid vignetting and have the beam

occupy the same solid angle (with the same orientation) at each point of the reflecting surface, the source or detector should be placed in the focal surface (or immediately behind a field stop in the

focal surface) of the primary optics, as shown in figure 7. Then the rays passing through each

point of the field stop form a parallel beam (focused at infinity) at the reflecting surface. But, if that beam is wide enough to intercept the entire reflecting surface at all angles, it must extend

beyond it in all cases, as shown. This means that the supports, housing, or whatever is adjacent to the defined reflecting-surface area will also be irradiated by the source and may contribute erroneous reflected radiation to the collected beam reaching the detector, to the extent that they

are not perfectly black or nonreflecting (configuration (3) of fig. 6). On the other hand, if the field stop is in the image plane rather than the focal plane, so that it is sharply imaged at the reflecting

surface, then rays will reach each point of that surface from the entire entrance pupU or aperture

25
OPTIC AXIS

/SOURCE OR RECEIVER ^ IN FOCAL PLANE

-APERTURE (ENTRANCE AND EXIT

PUPILS OF PRIMARY OPTICS SHOWN

SUPERIMPOSED FOR SIMPLICITY)

THESE RAYS CAN PRODUCE

UNWANTED REFLECTIONS OR

SCATTERING BY STRIKING

HOUSING.SUPPORTS, ETC.

FiGUBE 7. Simplified configuration when source (or receiver) is located in focal plane (or just behind field stop in focal plane) of primary optics.

26
of the primary optics and the solid angle of the beam will vary from point to point across the

reflecting surface, as shown for the extreme edges in figure 8 (this would be the case in configurations (1) and (2) of fig. 6).

-OPTIC AXIS

SOURCE OR RECEIVER

APERTURE (ENTRANCE AND EXIT

PUPILS OF PRIMARY OPTICS

SHOWN SUPERIMPOSED FOR

SIMPLICITY)

SAMPLE (REFLECTING SURFACE)

Figure 8. Simplified configuration with source (or receiver) and sample {reflecting surface) in conjugate image planes of primary optics

27
C. Sub-Surface Scattering (Edge Effects)

Earlier (sec. II. A), we treated sub-surface scattering in terms of the BSSRDF (eqs (1) and (2)) without making any assumptions about the mechanism(s) involved. However, it may be helpful to look briefly at a few considerations, without examining possible mechanisms in any great detail. For example, for very rough surfaces we may choose a reference plane lying just above the highest

points. Then microscopic analysis would reveal many interreflections between the surface

irregularities by which an incident ray would be returned as reflected radiation through a point of

the reference plane different from the point through which the incident ray entered the space

beneath. An example might be a roughened metal surface, or again, the surface of the moon, with

large hills and valleys, viewed from the great distance of the earth. Probably the most common

situation, however, is the body scattering within the material of most diffuse reflectors. For body scattering materials, besides the Fresnel reflection, there are diffuse reflections produced by internal scattering [22-25], usually multiple scattering, in the material below the

nominal reflecting surface (usually the boundary between that material and the air above it). This occurs with diffuse reflectors, which include, in varying degrees, nearly all natural reflectingsurface materials, depending to a great extent on their opacity, i.e., on the degree to which the

incident beam penetrates into the material before being absorbed or scattered back through the

surface again, each ray emerging at a point which, in most cases, is different from the point of

incidence (for that ray). The reflected rays each include radiation which is incident at adjacent

points as well, and for the more translucent materials, the separation between the points involved

in such interaction can be quite substantial [26]. In such cases, it is necessary to either irradiate a

large area and view a smaller area (or vice versa), as indicated in section II. A.

In order to take such interaction into account, we should refer back to eqs (1) and (2) where

the BSSRDF represents the basic property of the material. For uniform irradiance and a uniform

and isotropic sample surface, the expression for the reflected radiance is simplified to the relation expressed by eq (6) which is rewritten as:

dLr =dEi{di,(t)i;k)-\ S{di,(t>i;dr,4>r;r)-dAi (30)

where

r = [{Xi-xrf + iyi-yrf]^. (8)

The reflected radiant flux can be expressed as:

d^r =

I dLrdClrdAr

^ r

= dEi{di,(t)rM-dCtr-

\ I S{di,(}>i;er,(f>r;r)-dAi-dAr. (31)

JA, JAi

The inner integration can be written as

[

S{ei,(f>i;er,(l>r;r)-dAi = 277 [

"

S{dt,<l>t;er,<t>r;r)-r-dr (32)

•>A, •'O

where is the radius of the circle about {Xr,yr) beyond which increasing the size of the irradiated

sample has no further effect on Lr, i.e., beyond which there is no significant interaction between

dAr and dAi. In other words, S diminishes rapidly enough so that

277 f S{du(t>i;dr,<t>r;r)-r-dr^ 0. (33)

28
This means that, when the two areas Aj and Ar are unequal, with one completely contained within

the other and with a band of width greater than Tm between their margins all around, the integral over the larger area is effectively limited to just the area covered by the smaller plus a band of

width Fm surrounding it. If we denote the area of such a band, of width r^, surrounding the

smaller area as a{rm), then, if lAi+a{rm)] lies wholly within Ar, the double integral of eq (31) becomes

but if we reverse the directions of all rays so that incident and reflected beams are exactly

interchanged, then Ar+airm) lies wholly within yi, and the double integral becomes, instead.

Since, in this case S{di,(t>i;Qr'4^r',r) involves the coordinates (A;,,y,) and {Xr,'yr) only symmetrically

through r, it follows that the integrals (34) and (35) are equal. Hence, it is immaterial which of the areas is the smaller, the irradiated area /i, or the defined area Ar from which reflected radiation is collected and measured. For a given magnitude of that smaller area, as long as it is contained in the larger area and separated from the boundary of the larger area by a band of width equal to or

greater than in all directions, the reflected radiant power collected and measured will be the same. Furthermore, if Ai and Ar are concentric circular areas that can be adjusted in size, r„ can

be estimated or measured (with respect to the noise level or resolution limit of the reflectometer) by

starting with them nearly the same size and gradually increasing the larger one until there is no

longer any observable increase in the signal from the radiation detector. At this point the radial separation between their boundaries is equal to r^.

It is assumed throughout the foregoing discussion that the sample size Ag » Ai or Ar and

that the distance from the boundary of the larger area J, or Ar to the sample edges is much

greater than rm. Otherwise, there may be disturbing internal reflections and/or scattering from the

margins of the sample. It should also be reiterated that this analysis applies only to samples that have uniform isotropic scattering properties parallel to the nominal reflecting surface, and it should be noted that it is essential that the uniformly-irradiated sample area extend in all directions beyond the edges of the observed reflecting area, or vice versa, by more than the

distance r^, i.e., that r > for each pair of incident and exitent points lying, respectively, on the

margins of .4, and Ar.

It should be emphasized that this quantity is not a fixed distance for a given material; it is a function also of the desired degree of accuracy or approximation accepted in eq (33). An

example, which provides a method for directly estimating r^, is to visually (with suitable

precautions for eye protection) observe the reflection from a point intensely irradiated by a sharply

focused beam, such as that from a laser source. The radius of the visible band of diminishing

radiance or brightness surrounding the intense central spot is r^, to the degree of approximation

established by the strength of the source and the threshold of visibility for the particular

circumstances. For larger (>2rm across in any direction), sharply focused, uniformly illuminated

areas, the width of the band of gradually diminishing brightness surrounding the central region of

uniform brightness will be approximately 2rm. Note also that is usually spectrally dependent,

having different values for different wavelengths.

From the equality of integrals (34) and (35) it follows that the measurement configurations (1) and (2) of figure 6 are equivalent. In the case of configuration (1), the choice of may be

described as being large enough to collect all of the reflected light due to the irradiation of

while in the case of configuration (2), the choice of may be described, as before, as being large

enough to include all of the irradiated area which contributes significantly to the radiance from Arsr->] (34)

[m^-sr-']. (35)

29
Of the two, configuration (1) of figure 6 is the one that is more likely to be used in a reflectometer

that measures reflectance, rather than a reflectance factor, by measuring, separately, the radiant

flux in both the incident and reflected beams. On the other hand, configuration (3) of figure 6

should not be used for materials for which sub-surface scattering is appreciable, but rather should

be restricted to cases in which surface (Fresnel) reflectance of highly opaque material, such as metal, is to be measured. In this configuration, as well as in configurations (1) and (2) for which

is not chosen large enough, scattered radiation passes laterally beyond the edges of the defined area with no opportunity for any of it to be returned into the reflected beam, or there is not enough (if any) compensating scattered radiation coming back through those edges from irradiated

surface elements outside the defined area. These "edge losses" may account for otherwise

unexplained differences in reported values of reflectance or reflectance factor for a given sample

measured with different instruments, since the exact configurations employed in the different instruments vary considerably.

For materials that are uniform, but are non-isotropic, the illustrative situation of an intensely

illuminated point would no longer produce a circular region of gradually diminishing brightness.

Instead, it would be elliptical or perhaps even more irregular and 2rm would correspond to its greatest width (diameter) in any direction. Furthermore, the requirement for uniformity, that S be

independent of the location of the temporarily fixed point, can also be relaxed to a requirement for only statistical macroscopic uniformity, with variations only over small enough distances so that they are not significant.

D. Extrapolation to Different Geometries

It is easily shown that, for any beam geometry, the reflectance is given by

p(a),;Wr) = fr{(Di;a>r)'^r [dimensionless], (36)

where

fr((Or,<Or) ^ [Vinrilr)]-

I \ fr{du<i>i;er,<l>r)-d^r-dili [sr-»] (37)

is the averaged BRDF, with respect to projected solid angle, over both the sohd angle of incidence

and the solid angle of reflection (collection). In fact, this is readily apparent upon comparing eqs

(36) and (37) with eq (16). Thus, as the beam geometry changes, as long as the average value of

the BRDFfr is not significantly changed, it is only necessary to multiply it by the new value of fir,

if any, to obtain the reflectance p for the new beam geometry. If no measurements are available

that were taken with the beam geometry_for ivhich a reflectance value is required, the best

estimate is obtained by using the value of fr (based on a measured value of reflectance factor) for the beam geometry that most nearly approximates the desired one, and multiplying it by the

appropriate value of O^. If possible, also try to take into account any supplemental data that might

indicate how the value of fr may vary with the directions involved in the differences between the

two geometries. Some practical problems and some measurement results are reported by Hsia and

Richmond [27]. Part II of their paper [27] is a useful bibliography on diffuse reflectance.

E. Other Radiation Parameters and Fluorescence (or Phosphorescence)

Reflectance, like absorptance and transmittance, may be highly dependent on the wavelength

or frequency v [THz] (as distinguished from modulation or fluctuation frequency / << i^) of the

radiation. Accordingly, selective reflection may frequently alter the spectral distribution in a beam

of radiation, so that the spectral-total reflectance (all wavelengths), even of identical surfaces, may

change as a beam, which includes an extended distribution of wavelengths, undergoes successive

30
reflections. If there is interaction between spectral and geometrical parameters, as just noted as a

possibility in materials with significant internal scattering, the geometrical distribution may also be

affected. In so-called linear situations, where propagation effects, such as reflection, are independent of the strength of the beam, mathematical treatment is simplified by dealing with

relative factors and distributions, usually normalized to a maximum of one. We will not consider

non-linear situations where radiation effects are intensity-dependent except to remind the reader

that, with high-powered lasers becoming available, such non-linear optical effects are being

encountered more frequently as time passes. The bidirectional reflectance-distribution function to be used in evaluating all of the

reflectances, reflectance factors, and MRDF's can be expressed by the weighted average of the

BRDF with respect to wavelength, with the spectral distribution of the incident radiation beam as the weighting function. If, however, there is no interaction between the wavelength or frequency

dependence and the geometrical dependence of reflection, the functions are separable. Then

fr{ei,(t>i;dr,(i>r;k)=fr{di,4>r,dr,cf>r)-fr{>^} [sr"'], (38)

where fri^) is normalized to have a maximum value of one by choosing

fr{du(l>uOr,(}>r) ^ fr{6u4>i\erAi\K) [sr"'], (39)

where Ap is the wavelength at which the value of the right side of eq (39) is a peak or maximum;

hence fr{K>) — 1- Then it is only necessary to multiply any reflectance, reflectance factor, or MRDF, computed from the BRDF given by eq (39) by the weighted average of /r(A.), with the

spectral distribution (relative is sufficient) of the incident beam as the weighting function xx(A),

i.e., by

/rUx(A)]=f Xx(X)-/r(A)-rf\/| XK{k)-dk [dimcnsionlcss], (40)

-'0 0

where x\{k) = X\{K)/C is the relative spectral distribution of the incident beam; XkQ^) is the

spectral radiometric quantity characterizing the beam; X may be any one of the radiometric

quantities Q, ^, I, E, M, or L; C is a constant with respect to wavelength.

Unfortunately, we are not aware of any data that will establish the extent to which there may

be interaction between geometrical dependence and spectral dependence, except for the knowledge

that it is a significant factor in some internal-scattering situations, as pointed out earlier, in addition to the obvious case of a diffraction grating where there is clearly substantial interaction. Certainly, there will be some wavelength bands over which the interaction is so slight that this last procedure will yield a good approximation, and it will probably hold more widely. But until more

complete bidirectional reflectance-distribution data are gathered to establish such dependencies, we can only speculate and caution against the possibility of interaction effects. Of the radiation parameters [5,28], the modulation or fluctuation frequency / {«v) appears

to be unique in that, up to the limit of the frequency-response capability of available radiation

detectors, there ordinarily appears to be no frequency-/ dependence on the part of the propagation

properties of matter-radiation interaction, the transmittance and the reflectance (or scatterance).

This means that coding of an incident beam by time-modulation or chopping can be used, and is widely used, to distinguish between its effects and those due to other ambient radiation that is difficult to control and/or measure accurately and comprehensively. For example, in many

reflectometers, the incident beam is chopped, and synchronous amplifiers are used to insure that the receiver-amplifier is responding only to reflected radiation produced by that incident beam and

not to reflected radiation from other ambient incident rays and/or emitted thermal radiation from

the sample surface and from other ambient objects, including the collecting optics, if any, of the

receiver.

31
Finally, the last of the radiation parameters, polarization, is often neglected or ignored, and we know of no published treatment that adequately deals with its implications for radiometry,

although it can strongly affect the propagance, particularly the reflectance, in many situations. Chapter 6 on polarization has been prepared for the "Self-Study Manual on Optical Radiation

Measurements" [28] and has been published in June 1977 as NBS Technical Note 910-3, and we

hope that it will help to clear up some of the problems in this area. Meanwhile, we note that a full description of the polarization of a beam can be given in terms of the four Stokes parameters [29],

each of which can be regarded as representing a component of that beam. Then, with four such

components for the incident beam, each possibly contributing to four such components in the

reflected beam, the most general description of the possible interactions would require a matrix of

sixteen (four x four) BRDPs. This matrix approach, but with fewer components (not the most

general case), is used in [30]. This topic will not be developed further here except to re-emphasize the fact that polarization can have a very large effect on reflectance, particularly in the case of specular reflectance. This

may not be particularly significant when measuring a single reflection of unpolarized incident

radiation with an unpolarized detector (one insensitive to polarization). Too often, however, it is assumed that an incident beam is unpolarized when, in fact, it is not, and/or that a detector is unpolarized when it is not. Actually, any off-normal emission [31], or off-normal reflection or

refraction of an unpolarized incident beam, will produce some polarization. The latter, however, is often obscured because the cylindrical symmetry of refraction by circular lenses produces no net

polarization. Similarly, there is usually no net polarization from circular focusing mirrors used onaxis, as in a Cassegrain configuration, but off-axis or asymetrical systems, such as a Newtonian

configuration or an off-axis paraboloid, may exhibit substantial polarization characteristics.

Fluorescence (or phosphorescence), if present, greatly complicates the measurement of

reflectance. The phenomenon of fluorescence involves the absorption of photons within one

frequency (photon-energy) range to produce emission of photons within a second, lower frequency

(photon-energy) range after a time interval governed by an exponential decay, characteristic of the

half-life of the excited atoms. If such re-emission persists noticeably (sometimes even for hours) it

is called phosphorescence.'' Reflection, on the other hand, is the process by which radiant flux incident on a surface or medium leaves that surface or medium on the incident side without

change in frequency (spectral frequency p = c/Aq)- Thus, the flux produced by fluorescence (or

phosphorescence) is, by definition, not reflected flux. But both will be present in any real measurements of fluorescent (or phosphorescent) materials so that spectral dispersion of both

incident and observed radiation is required to distinguish between them. If the incident beam is restricted to a narrow spectral range by a monochromator, radiation of that same spectral

frequency, observed to be leaving the surface only when it is irradiated, is reflected radiation.

Similar observations of a lower exitent spectral frequency establish the presence of fluorescent (or phosphorescent) emission. Further, if the incident beam is modulated, the fluorescent (or

phosphorescent) emission will be found to be modulation-frequency dependent because of the time

constant associated with the exponential decay characteristic of the excited atoms. The details of

such measurements are a separate problem that will not be discussed further here.

F. Use of Reference Standards

The absolute value of reflectance is usually very difficult to measure. The main problems are

to measure the solid angle of viewing and to measure the flux in the incident beam accurately.

This usually requires removal of the sample of reflecting surface as well as shifting of the detector

(collector) and it is hard to be sure that the geometry (throughput) of the measured beam is exactly

the same as that of the beam incident on the sample (when it is replaced) that produces the

"Fluorescence is characterized by a time delay of less than about 10 [ns]. Re-emission after 10 [ns], sometimes much later than 10 [ns], is called phosphorescence.

32
reflected beam. For precise and accurate measurements, the flux in the incident beam and its distribution in position and direction, as it impinges on the sample surface, must be completely

known. (We ignore the equally important spectral and polarization parameters for this discussion.)

However, once the absolute reflectances of a set of reference standards are carefully measured,

the reflectance measurement of an unknown sample can be more easily done by the comparison

method. The BRDF of the chosen reference standard should resemble that of the sample as closely as possible to minimize the sensitivity to instrument-alignment errors. On the other hand, the measurement of a reflectance factor being inherently a comparison

measurement, the beam-defining apertures for both source and detector (receiver) beams can be

left undisturbed while a sample and a comparison-standard surface are interchanged. Accordingly,

with identical beam geometry thus insured, it is much easier to achieve high precision (repeatability)

in the measurement of reflectance factors. The main limitation on measurement accuracy is then

the fact that there is no ideal (100%) perfectly diffusing (lambertian) comparison-standard surface. When the departure of the actual standard from 100 percent reflectance, and from a perfectly

isotropic (lambertian) diffusing characteristic, are accurately known, appropriate allowance can be

made for them. For the highest accuracy, however, this also involves the exact beam configuration

and throughput for both the incident and reflected beams and the BRDF of the comparisonstandard surface (in order to know exactly how much it differs from the ideal perfectly diffusing

standard in the particular measurement configuration). Finally, a knowledge of at least the relative BRDF of the unknown surface being measured is also needed for any estimate of the accuracy and

signficance of the final measurement result. Only with this information can the sensitivity of the

measurement result to small changes in beam geometry be assessed. The final result, of course, is valid only for the particular beam geometry of the measurement. The BRDF information is needed

to determine how restrictive that limitation may be in the particular case.

Incidentally, it turns out that one of the comparison standards which can be most accurately

characterized in terms of its BRDF for such use in the measurement of reflectance factors is a specular reflector. For convenience, to keep the total incident flux on the detector (receiver) more

nearly at the same level for all of the measurements, one of low reflectance, such as a polished

flat black glass of known specular reflectance Psp is used. Its BRDF is then closely approximated

byeq(ClO).

V. Summary, Conclusions, and Recommendations

Within the domain of geometrical (ray) optics, the description and specification of the

geometrical reflecting properties of any surface are given by the bidirectional scattering-surface

reflectance-distribution function (BSSRDF) S. In most cases, however (see sec. II. A for the

necessary conditions), they can be adequately specified by the simpler bidirectional reflectancedistribution function (BRDF) fr. Other useful geometrical reflectance quanfifies, including reflec- tances and reflectance factors, are expressed and interrelated in terms of the BRDF. The effects of radiation parameters, other than geometrical, including polarization, and of fluorescence, are

briefly discussed and their importance in the measurement and specification of reflectance

quantifies is indicated.

Within the limited scope we impose on ourselves for this monograph, our principal

recommendafions for the approach to the geometrical description and specification of reflecting

properties of surfaces, and for nomenclature to facilitate that approach, are:

1. The term "bidirectional reflectance-distribution function" (BRDF) and symbol /r(^,,</),;6r,0r) should be adopted and this quantity, under appropriate conditions (see sec. II. A), should be

recognized as the basic quantity to be used in defining all other reflectance quantities or measures of reflectance, except in some situafions involving substantial sub-surface scattering.

2. The beam geometry should always be completely specified (including uncertainties) in any

quantitative statement about reflectance.
3. The nomenclature of tables 1 and 2, including McCamy's notation for right-circular-conical

beams, should be adopted.

4. Other schemes of concepts and nomenclature to cover special beam geometries should be

considered acceptable as long as they are clearly defined in terms of bidirectional reflectancedistribution function (BDRF) or the bidirectional scattering-surface reflectance-distribution

function (BSSRDF) (including adequate consideration of any special conditions).

With the foregoing as a point of departure, we hope that, in time, our approach can be

extended to also take into account the effects of the other radiation parameters, particularly

polarization.

While we strongly urge the appropriate organizations concerned with establishing nomenclature

standards to consider the foregoing recommendations, we should, at the same time, clarify our

position with respect to all such standards. Standardization is an important requirement for

efficient communication. Standards should be adopted as rapidly as a workable consensus in support of them can be established. On the other hand, they should never be permitted to become

a strait jacket that stifles innovation and progress. An author who feels that he needs new and

different nomenclature to adequately express his ideas should always be free to adopt and use such nomenclature. However, he should always do so explicitly, defining his nomenclature or usage clearly and, as far as possible, showing how it is related to, or differs from, any pertinent

standard nomenclature. In fact, we fully expect that we, ourselves, may not infrequently find occasion to depart from the detailed nomenclature proposed in this monograph. But we also accept

the obligation, when doing so, to define any such nomenclature and to relate it, where appropriate,

to existing standards and/or to our recommended standards.

In conclusion, we want to emphasize that this monograph is not primarily intended to propose

or advocate any particular nomenclature. Far more important than the nomenclature are the

underlying concepts which we believe are essential to full understanding and effective treatment of

reflectances and related quantities. We introduce nomenclature only in order to be able to talk about the concepts. The essential point is the recognition that complete specification of the

reflecting properties of a highly opaque material'^ must take into account the directional

distribution (in terms of radiance) of radiation in both the incident and reflected beams. Finally,

when there is significant internal scattering, which is true in some degree for all diffusely

reflecting materials, it must also be recognized that rays incident at one point contribute to the

exitent "reflected" radiation from adjacent points. We have shown how these considerations can

be treated and have proposed nomenclature to facilitate that treatment.

VI. Appendix A. Background and Basis for Recommendations

In this appendix we present, briefly, the basis for our recommendations, particularly where

they represent changes from earlier proposals and practices.

Reflectance nomenclature, primarily from the standpoint of beam geometry, including

concepts, terms, and symbols, was discussed by Nicodemus [10] in an analysis of the interrelation- ship between directional reflectance and directional emissivity or emittance (Kirchhoffs law), and

by Judd [12] from the standpoint of reflectometry. Subsequent experience revealed inadequacies

in and inconsistencies between the two approaches, as well as useful interrelationships. Within a year following the publication of Judd's paper [12], four of the authors of this monograph

(Nicodemus, Richmond, Ginsberg and Limperis) had met and outlined most of what is presented

here. We have tried to eliminate the inadequacies and inconsistencies of the two earlier papers, recommending nomenclature that incorporates the best from both of them. This is supplemented

by the multivariate approach (see table 3), devised by two of us (Ginsberg and Limperis) and by

" Material in which there is neghgible separation (for a particular application) between the point of incidence of a ray and the points of exitence of all resulting internally scattered rays leaving the "reflecting" surface.

34
the approach of McCamy [13] who was concerned with transmittance and reflectance in connection with photographic imagery. Preparation of this paper for pubHcation was repeatedly

delayed for a variety of reasons. Meanwhile, Nicodemus became aware that a number of his earlier recommendations for terminology and symbols, that we had agreed were undesirable and in need of revision, were being rather extensively used. Accordingly, he published a letter to the

editor of Applied Optics [9] to set the record straight on those points.

The changes from Nicodemus' earlier nomenclature [10] included in our recommendations are summarized in table 4, from [9]. The reasons for them are also essentially those given in [9].

Table 4. Comparison between recommended reflectance nomenclature and that of [10]*

Item No.

Quantity,

Recommended terminology

Quantity,

Ref-[10] terminology

Recommended

symbol

Ref-[10] symbol

1 Radiant power P

2 Radiant intensity / J

3 Radiant exitance (radiant emittance) M W

4 Irradiance E H

5 Radiance L N

6 Solid angle 0} n

7 Projected solid angle n ft' 8 Bidirectional reflectance- (Partial reflectance, or fr p' distribution function bidirectional reflectance) 9 Directional-hemispherical (Directional reflectance) /"*(»,-,</),) reflectance 10 Hemispherical-directional

reflectance factor (Directional reflectance) R(2TT-dr,<t>r)

from [9].

Items 1 through 5 merely establish conformity with the radiometric nomenclature of [4]. Items

6 and 7 follow a practice suggested by R. Clark Jones [32], leaving the prime notation available for other purposes. Solid angle and projected solid angle are explicitly defined and discussed in [2]. Item 8, the bidirectional reflectance-distribution function (BRDF) fr, is the most important

entry in table 4. Except for considerations of sub-surface scattering (see the last paragraph of this section), it is the basic quantity, underlying all of the reflectance nomenclature recommended in this paper. It has [2] unit-dimension [sr"'] and its value may range from zero to extremely high

values, even going to infinity in the delta-function form for ideally specular reflectance in eq (CIO).

It is, therefore, important to distinguish it clearly from reflectance, the dimensionless ratio 0 :£ p 1 of eq (15a). However, both the terms "partial reflectance" and "bidirectional reflectance"

(especially the latter) and the symbol p', to denote the BRDF, have caused much confusion by

failing to emphasize this distinction adequately. The poor choice of terms is probably most to blame, since the prime notation is widely used for derivatives elsewhere in other connections.

However, in line with our desire to free the prime for other purposes and because the confusion

with p has occurred, even though the terminology rather than the notation may have been

primarily responsible, we are recommending the notation fr. Furthermore, both the new term and new symbol help to emphasize the nature of this quantity as a distribution function, encouraging

appropriate utilization of existing mathematical treatments of distribution functions in general.

Finally, the fact that two directions are involved is clearly denoted by the physically descriptive

modifier "bidirectional" and by the explicit statement of functional dependence in the full notation

fr{eu<t>i-A,<i>r). Items 9 and 10 clarify the nature of two quantities (both called reflectances in [10]) by the

application of Judd's (modified) terminology and symbols, which show that while one is a

reflectance, the other is actually a reflectance (radiance) factor. As shown in [10], and recognized

in [12], these two quantities for a given direction are always equal (i.e., p{d,4>;2TT) — R{2n;9 ,4))) as a direct consequence of Kirchhofrs law (as orginally stated in directional form), regardless of

Helmholtz reciprocity [9].

35
Judd's [12] scheme of functional notation for designating the beam geometry for any

reflectance quantity has been adopted with certain modifications. We prefer the semicolon used in an earlier draft of Judd's paper [12], rather than the colon that appeared in the published paper,

for separating the parameters designating the incident-beam geometry from those designating the

geometry of the reflected (collected) beam. This avoids possible confusion with similar use of a colon in other notations. We also prefer the more explicit (d rather than Judd's g as the symbol for the conical beam geometry, and have extended it to include, also, the extremes of an element of

solid angle do) (which we use interchangeably with {d,(j)) to denote a well-coUimated beam in a given direction) or a fuU hemisphere of 2tt [sr]. The use of w is also more consistent with the other

symbols; this way, all three

den, (o, and 2tt—represent solid angles. The nine permutations of

three incident and three reflected (collected) beam geometries (see fig. 3) used by Judd have been

followed in tables 1 and 2, except that the order has been reversed. We start with the bidirectional

quantity and end with the bihemispherical quantity. This seems more consistent with the approach

of building up all of the quantities by integration and/or averaging of the BRDF, as shown in the

last column of each table. In table 1, note that three quantities—bidirectional reflectance c?p(0,,(^,;0r,</>r)» conicaldirectional reflectance dp{(jii\,dr,<i>r), and hemispherical-directional reflectance dp{27r;dr, <t>r)—are

all designated as differentials by the notation used, the reason being clear from the last expression on the same line, giving each quantity in terms of the BRDF. Judd [12] speaks of integrating the

bidirectional reflectance to obtain other forms of reflectance but, unfortunately, used the notation

p rather than the differential notation dp for all three of these quantities. All of Judd's [12] rather

involved sets of interrelationships between the various quantities in both tables 1 and 2 are readily

apparent and derivable from the consistent set of expressions for each quantity in terms of the

BRDFfri6i,4>i;6r,(l>r) in the last column of each table. In table 2, we propose a clear departure from the terms and symbols of Judd [12] and of ANSI

[4] and its source, the CIE [14], by abandoning the term radiance factor and the symbol 13 for the

three quantities: bidirectional-reflectance factor /? (^, ,</>,; 0r,</>r), conical-directional reflectance factor

R{(t)i;6r,4>r), and hemispherical-directional reflectance factor R(2Tr;dr,4>r)- In this way, the close

interrelationship of all nine quantities in table 2, all of which are contained in the expression for the biconical reflectance factor R{(Oi;(Dr), if the range for the solid angles w, and is extended to include the extremes of an element of solid angle doj and a full hemisphere of 27t [sr], is more

clearly apparent. On the other hand, since first deciding on this recommendation, we have

realized that there are occasions when one may wish to make a general statement about a

radiance factor (i which applies equally to any one of the three, or about the radiance factors /3 generally, which applies to all three, but in neither case to any of the other reflectance factors. In the newly proposed nomenclature, it is very awkward to make such a distinction between these

three quantities and the other reflectance factors. Accordingly, we also favor retention of the older

nomenclature as an acceptable alternate form.

From the outset, we were aware, also, of the phenomenon of sub-surface scattering and so- • called "edge effects" in diffuse reflectors, particularly in obviously translucent materials.

Accordingly, the discussion of the "edge effects" (sec. IV. C) was developed. However, since it justified wide continued use of the formalism already established on the basis of the BRDF, we

' left it there and made relatively little mention of it earlier. Later, during reviews at NBS, in order

to improve clarity and continuity, Venable and Hsia proposed a new approach in which the more

general BSSRDF was introduced first and the BRDF was then developed as the important special case. Meanwhile, their publication [8] established the even more general scattering function S, of

which the BSSRDF is a special case. The main body of this monograph was, accordingly,

rewritten using this approach.
VII. Appendix B. Details of Reflectance Derivations (A Thought

Experiment)

Consider a large, opaque hemisphere with walls and base of cold, rough, black material that does not transmit, emit, or reflect any radiation. Of course there is no real material that can fully

meet these conditions; but it is useful, conceptually, to postulate them in order to isolate and

clarify the concepts involved in the reflectance relations to be analyzed.

An element of completely opaque reflecting surface dA is positioned at the center of the base

and lies in the plane of the base. Here, too, we postulate a material so opaque that there is no

penetration at all, a condition only approximated by real materials (e.g., metals).'^ Outside the

hemisphere, there exists a field of radiation (radiant energy) flowing in all directions. However, the

hemisphere is so large (relative to dA) that, when holes are cut in the spherical surface to admit

this radiation from outside, only radially-directed rays of radiance L,(0,,</), ) [W cm"^ -sr"'] can

reach dA; all others strike the cold black walls and base and are absorbed. (This use of

differentials, such as the area element dA, and the relations of the calculus, based on assumptions

of continuity in the domain of geometrical optics, is well explained by Jones [7] and is also discussed in [2].) A small hole in the spherical wall, which subtends an elementary solid angle da>, at dA,

admits an elementary beam which strikes dA with an irradiance (see chapter 4 of [28], stUl in preparation)

dEi(di,(t>i) = LiidM-cosdi-dcji = LAOi,<f>i)-dVLi [W-m-^]. (Bl)

In radiometry, irradiance E [W • m~^ ] is not ordinarily treated as a directional quantity, except in a

vectorial method for lighting calculations in illumination engineering [33,34]. However, the

differential element dE is clearly a directional quantity as it is used here in eq (Bl). The incident

radiant flux or power reaching dA is then

d'Pi = dEi(di,ct>i)-dA [W]. (B2)

Note that, for generality, no restriction is placed on the incident radiance Li which may vary

with direction as Li{di,4>i), a possibility that must be recognized in dealing with real situations. However, as soon as we consider the effects of increasing the angular size cu, of the incident

beam, it becomes apparent that the reflected radiant flux may depend not only on the amount

contained in the incident beam but also on the way in which it is distributed with respect to incident direction within that beam. Only in the case of a perfectly uniform and isotropic diffusing

surface, with the same reflecting properties in all directions, will the reflected flux depend only on

the amount of incident flux and not on its directional distribution. For useful reflectance quantities, to characterize the reflecting properties of a surface for different beam geometries, the possibilities become prohibitively complex if we also try to take

into account different directional distributions of flux within the incident beam. The situation is greatly simpHfied, and the results are still of substantial utility, by assuming isotropic incident

radiance L; (isotropic for all directions within the defined solid angle Wj); hence Li is shown as having everywhere the same value outside the hemisphere in figure 9. Wherever feasible,

however, we retain the variable Li{di,4>i) for greater generality. In general, the incident radiant flux is either absorbed or is reflected into all directions within

the hemisphere by the surface element, as indicated by the arrows of varying lengths dLr{&r,4>r)

diverging from dA in figure 9. As noted in [5] and [10], even when the reflecting surface is highly

specular, there is always some, perhaps very small, scattering or diffuse reflection in directions

other than the specular angle. (Special considerations concerning such glossy or highly specular

'^The conditions underlying the definition of BRDF and the discussion of section IV. C should be noted, particularly for materials in which a substantial part of the reflection is due to sub surface-scattering.

37
surfaces are discussed in appendix C.) Often, however, we are not interested in all of this reflected radiation but only in that reflected into some specified solid angle a>r occupying only a

portion of the full hemisphere. In the present case, it is only into an elementary solid angle d(Or in a single direction {Or,(t>r), defined by cutting another small hole in the hemispherical wall and

placing a receiver R just outside to respond to the reflected radiation reaching it from dA through

the hole and to block any outside radiation from entering, as illustrated in figure 9.

Figure 9. ''''Thought-experiment"configuration for bidirectional reflectance.

38
The reflected radiant flux reaching the receiver through the element of solid angle datr is

d^r = dLr{6r,4>r) dClr dA = dLridr,<f>r)- COsdr ' duir dA

= dMr(er,<t>r)dA [W], (B3)

where dMr{Br',''t>r\ like dEi{di,(f>i) in eq (Bl), is a directional quantity, defined by (see chapter 4 of

[28], still in preparation):

dMAdr,<l>r)=dLridr,<t>r)dnr =dLr(er,<f>r)COSerda)r [Wm-^]; (B4)

Then, combining the basic definition of reflectance with eq (Bl), (B2), (B3), and (B4), the

bidirectional reflectance is

dpiei,<t>r,er,<}>r)=d<^r/d<Pi = dMr{er,<t>r)/dEdeM

= dLr{er,<i>r)-d^rldEdeiM = dLr{6r,<t>r)dilr/Li{di,(f>i)-d^li [dimensionfess]. (B5)

Now consider what will happen to the reflected radiance dLr and the bidirectional reflectance

dp as we change conditions to produce small changes in each of the factors on the right-hand side

of eq (B5). First, changes in the magnitude of the incident radiance L((0(,<|>() will be seen to produce corresponding proportionate changes in the reflected radiance dLr, leaving the ratio dp

unchanged. (We are not concerned, here, with the non-linear effects that may occur at extremely

high values of Lj , found only in the radiation beams from the highest-powered lasers.) Similarly,

smaU variations in the incident projected solid angle element d^lt (changing the size of the small

hole through which Li enters) will also produce proportionate changes in dLr, again leaving dp

unchanged. (At small values of 6, where cos^ == 1, the difference between dO and d(o may not be

perceptible, but as 6 approaches 7r/2 [rad], where cosd changes more rapidly with changing d, it will be clear that the direct proportionality is between dLr and rfftj = cosdi-dwt, not just rfoij.) Thus, dLr is seen to be a function of incident irradiance dEi = Li{di,(t>i)'d^li and, for a given value

of dEi, it wiU depend both on the direction {Oi,<l>i) from which that radiation is incident on dA and

on the direction {6r,(f>r) in which it is reflected (and collected), as can be established by shifting the

locations of the holes in the hemispherical shell. Hence, we will write dLr{Bi,4>i;dr,<f>r;Ei), as in eq

(9), and we recognize that this quantity, since it is produced by the incident radiation interacting

with the reflecting surface dA, is a dependent variable that cannot be independendy varied. It changes in direct proportion to changes in the denominator on the right-hand side of eq (B5),

leaving the ratio dp unchanged. However, dilr can be varied arbitrarily to produce directly

proportionate changes in d^r, the reflected flux that is collected and measured, and hence in dp.

Thus, dp is not a measure of just the reflecting properties of the surface for the given pair of

directions but is also directly proportioned to the size of the element of projected solid angle of

reflection (and collection) dClr . The invariant quantity for a particular element of reflecting sjarface

dA, the one that characterizes its (geometrical) reflecting properties, is the bidirection^ reflectancedistribution function (BRDF) of eq (7):

fr(ei,<i>i;Br,i^r) = dp{ei,4>i;er,(f>r;i^r)ld(lr = dLr{eu<l>i;er,<f>r;Ei)/dEi{dt,(f>i) [sr"*]. (B6)

39
This quantity, given by eq (B6), is unaffected by small changes in the geometry of the incident or the reflected elementary beams, as long as they are still elements in essentially the single

directions indicated, or by changes in the strength of the incident radiation. In other words, it characterizes the reflecting property of the surface element dA for the given pair of directions.

Furthermore, by Helmholtz reciprocity, which holds in the absence of polarization and magnetic

fields, it is immaterial which direction, of the pair, is that of the incident radiation:

m,</>i;^2,02) =m>2;e.,<^>i) [sr-']. (B7)

Consider, next, what will happen to the reflectance as the projected solid angle of the

reflected beam is made much larger, so that dftr in eq (B6) increases to become ilr, with

correspondingly finite (i)r, as illustrated in figure 10. The incident radiant flux is still given by eq

(B2). The reflected flux, however, is now given by

d<Pr=dA \ dLr{e„(t>i;dr,<l>r;Ei) dn, [W]. (B8)

dA

Figure 10. "Thought-experiment" configuration for directional-conical reflectance.

In order to relate this to the incident radiation, note that eq (B6) provides the relationship between

the incident radiation and the reflected radiance dLr in every element of solid angle do)r within the

solid angle a>r. Accordingly, combining eqs (B6) and (B8),

d<Pr = dA dEt(dM ( Meu<t>i',er,4>r) d^r [W]. (B9)

40
The directional-conical reflectance is then obtained by combining eqs (15), (B2), and (B9):

p{et,<t>i;ojr) = d^r/d^t

=

I fr{'^i,<l>i',dr,<l>r)'dClr [dimensionless], (BIO)

in agreement with table 1. The same result can be obtained quite directly by rearranging and

integrating the first line of eq (B6):

PiOi,<t>i;(^r) ^ dp{di,<t>i;dr,(t>r)

—

I fr{Qii4>i'-,^ri't>r)'d^r [dimcnsionlcss]. (BlOa)

However, we presented the first derivation because this second approach, although mathematically

simpler, may leave some doubt whether the mathematics adequately represents, the physics of the

situation. Now, returning to the original configuration with two small holes, consider, instead, what

happens if the hole for the incident beam is enlarged substantially, so that dH, in eqs (Bl) and (B5)

is increased to O,, with corresponding finite cd,. As depicted in figure 11, the resulting reflected radiance Lr{dr-,4>r) is made up of contributions (see eq. (B6))

dLr(du(t>i;dr,<f>r;Ei) ^fr{di,<lii-A,4>r)-dEi(dM [W • m'^^ • sr" ' ] (BID
for each

dEi{diM=Li{eiA)-dili = Li{ei,(t>i)-cosdi-d<Oi [W-m-^] (Bl)

for every element of solid angle rfw, in a direction {Bi,<j>i) throughout the entire solid angle Wj,

corresponding to the projected solid angle

Vli= I cosdi'dcoj = I I cos6i-s'mdi-ddi-d^i [sr]. (B12)

The total reflected radiance is then obtained by integrating eq (Bll) over the solid angle of

incidence w, :

LAQr,<^r) =

I dLriei,ct,r,0r,(l>r-^i)

= 1 fr{ei,(f>r,0r,4>r)-dEiieM

=

I Li{du<i>i)-fr{eu4>i-A,<i>r)-d^i [W-m-^-sr-]. (B13)

Accordingly, the reflected flux is given by

d<Pr ^ Lridr;())r)-dflr-dA

= dilr-dA\ LdeM-fr{Qi,^i;Qr,<t>r)-d^i [W]. (B14)

The incident flux is obtained by integrating eq (Bl) over the solid angle of incidence at, and

multiplying by dA:

d^i=dA-\ dEi(di,4>i) = dA - I Li{du(i)i)-dCLi [W]. (B15)

The conical-directional reflectance, for the geometry depicted in figure 11, is the ratio of eq (B14)

to eq (B15):

dp((iii;6r,<l>r) = d^rld<^i

^dVLr-\ Ldeu4>i)-fr(di,<iii-dr,^r)-d^j\ Udi,cl),) -dil,

[dimensionless]. (B16)

The expression in eq (B16) is the general expression, where no restriction has been placed on

Li(di,(t>i). Strictly, then, this should be termed the conical-directional reflectance for non-isotropic

incident radiation. When the term conical-directional reflectance is used alone as in table 1, it refers to the simpler situation where the incident radiant is isotropic, i.e., Z,, is constant, within

the solid angle of incidence Wj. With this assumption, the constant term L, can be brought outside

the integral sign in both numerator and denominator in eq (B16), where it will cancel out, leaving

the conical-directional reflectance

42
dp{(Oi;6rj<t>r) = (dflr/i^i)' I fr{0i^<l>i;0r ^<f>r) ' dili [dimensionless], (B17)

again in agreement with table 1. From eqs (BIO) and (B17) it can be seen that, over a finite solid angle of reflection , the value

of reflectance is obtained by integrating the bidirectional reflectance-distribution function

fr{di,<t>i',^r,^r) with respcct to projected solid angle dCtr over that soHd angl6 cu^ and, over a finite solid angle of incidence oj;, it is obtained by averaging /r(^,,</)i;0r,<Ar) with respect to projected solid angle rfH, over that solid angle w,. We can, then, combine these two operations to obtain the

general expression for the biconical reflectance

p(Wi;ajr) = (l/fii)

1 fr {6i,4)i;dr,(l>r)-dClr-dClj [dimensionless]. (B18)

As previously pointed out, this is the completely general expression for all geometries if we

assume that the soHd angle doj < (o < 2tt [sr] for both w, and 0)^. Hence, all of the expressions in table 1 may be derived from eq (B18) by substituting the corresponding values for these solid angles. Note, however, this is not completely general, in the sense that it is based on the

assumption of isotropic irradiance within the solid angle of incidence oj,. The more general

expression, for non-isotropic incidence, is that given earlier in eq (15a).

VIII. Appendix C. Perfectly Diffuse and Perfectly Specular Reflectances

Perfectly Diffuse Reflectance

A perfectly diffuse or "lambertian" surface element dA is one for which the reflected

radiance is isotropic so that Lr is a constant, with the same value for all directions idr,(t>r), regardless of how it is irradiated. From eq (B13) it is apparent that this is possible only when fr =

fr,ci a constant, so that

Lr.d=fr.<i\ U{ei,4>i)-d^li=f,a-Ei [W-m-==-sr-i]; (Cl)

hence

fr.a=Lr.JEt [sr->]. (C2)

In terms of the perfectly diffuse reflectance Pd, which is the fraction of the total incident flux that is reflected isotropically (iso-radiance rays) in all directions (into the full hemisphere above the

element dA of the reflecting surface), we have

Pd{o>i;2Tr) = d^r/d^i = {Lr.d dA [ dClr) /l{E, dA)

— fr,di'Si,^i',^r,^r)"^ [dimenslonless]; (C3)

hence

fr.d{Oi,<t>i;dr,(t>r) = Pdi(Oi;27T)/TT [sr

43

(C4)
and

Lr.Aeu4>rA,(l>r)=pA(^i;27T)-Ei/iT [W-m-2-sr-»], (C5)

where

Pdid(i)i;2TT) = pd(ftj,;27r) = P(,(27r;27r) - a constant. (C6)

An ideal (lossless) diffuse standard reflector returns all of the incident flux so that

fJd = Pid - 1 (C7)

and

fr.idiOi,(t>uOr,c}>r) = V^T [sr"!]. (C8)

Rerfectly Specular (Mirror-Like) Reflectance

A perfectly specular (mirror-like) or regular reflecting element dA is one for which each ray of

incident flux produces only a corresponding reflected ray in the specular direction (Or = dt; 4>r —

4>i ± tt) [rad], so that

Lr.s„(dr,ct>r)=Ps,.-Lder,(t>r±7T) [W • m'^ • Sr'^ ]. (C9)

If the general expression for reflected radiance in eq (B13) is to include specular reflection, we

must find an expression for fr,si, which, when substituted into eq (B13) reduces it to eq (C9). This means that the integral must vanish (=0) for all values of 6r i= di and of <^r ^ <l>i ± tt [rad] and

must have a finite singularity when 6r = Oj and (t>r = *i>i — tt [rad]. This is not possible with any

normal functional form of BRDF fr but these conditions are satisfied by the Dirac delta-function

form

/r.«>(eM0,;er,cAr) = 2p,„-8(sin26»,-sin2ei,) •&((/),-(/), ±7r) [sr-'], (CIO)

where the Dirac delta-functions are defined by

S(u) = Oforu 9fc 0,

J8(u)-du = 1, and (Cll) ^f{u)-b{u-a)-du ^ f{a),

when the integration is carried out over any range of the variable that includes the zero of the

argument of the 8-function [21]. This is readily verified by making the substitution and carrying out the integrations in accordance with the defining relations for the Dirac delta-function, eq (Cll).

An ideal (lossless) specular reflector is one which also reflects all of the incident flux without

attenuation so that p,sp = 1 and

/M.p(^i,</)i;erA) = 2-8(sin2e,-sin20,)-8(</>r-</>,±7r) [sr"']. (C12)

Specular Reflection

Traditionally, specular reflection and diffuse reflection have been treated quite separately in ways that do not adequately recognize the broad spectrum of intermediate conditions filling, in

44
reality, the entire region between the two conceptual extremes of perfectly specular and perfectly

diffuse reflectance as well as a much broader region not on a direct line between (not just a mixture of perfectly specular and perfectly diffuse reflectance—see fig. 5). The distributionfunction approach presented here provides a much better means for modeling that reality and, in our

zeal to emphasize this, we have so far neglected some very real problems that arise in dealing with

practical cases of specular reflection. We have defined perfect specular reflectance as the condition where the reflected radiance in any direction Lr{dr,4>r) is directly proportional only to the incident radiance from the specular

direction Li{dr,4*r—'^)-, so that Lr{Or,<f>r) = Ps;)'^i(^r»</'r±^)- That is a perfectly good definition of

the ideal concept, but it is not a convenient form for dealing with practical specular reflectors that only approximate the ideal condition. For that purpose, it is more useful to consider that a specular reflector is one where the directional-hemispherical reflectance for a given direction of

incidence is equal to the directional-conical reflectance for the same direction of incidence where

the solid angle of reflection (collection) is a right circular cone of arbitrarily small vertex angle with

its axis oriented in the specular direction. In the ideal case, the vertex angle approaches zero as a

limit so that, using McCamy's notation [sec. III. A] for ot^.

Psp{0i,<t>i;27T) = Psp(^,,(/>i;^or = ^i,</>or = </>i±7r,Kr) [dimcnsionlcss] . (C13)

For real specular reflectors (see fig. 5), the equality is only approximate, since there is always

some, perhaps extremely little, scattering into other directions. The value of for which the

closest approximation holds true may become very small but will never actually reach the zero

limit for obvious reasons. Some of those reasons are mentioned briefly in commenting on figure 5 and in discussing glossimetry practices in section III.C. In any case of specular reflection, the value of Psi> in eq (CIO) is that which satisfies eq (C13).

Hence, it should probably be written as Ps;<(^M</>i) in eq (CIO), since it may vary with the direction

of incidence.

Often, a reflectance-factor measurement is not only the simplest but is also the most useful measurement of the reflecting properties of a sample. It can readily be extrapolated to estimate

the reflectance for beam geometries other than that used in the measurement, as long as the

average value of/r remains roughly the same over all of the directions involved. Clearly, however,

this comment applies only to fairly diffuse reflectors and does not deal with glints and specularities,

where fr changes very rapidly with direction. As a matter of fact, in the ideal case, the directional- conical reflectance factor corresponding to eq (C13) becomes infinitely large as k approaches zero. With specular reflectors, the need for extrapolation to different beam geometries is not so

likely to arise. When used as specular reflectors, e.g., for imaging, the significant reflected flux is mainly that reflected in the specular direction; when measurements are made it is only in that direction that there is likely to be enough reflected flux to make a measurement; and from eq

(CIS), it is apparent that a measurement of Psp wUl not be very sensitive to the size of a»r (as long

as (jJr > w,). Nevertheless, in evaluating the imaging quality of good specularly-reflecting optical

components, it is just the small inequality in the approximation of eq (C13) that is significant. It is the difference between the directional-heitiispherical reflectance and the specular reflectance, the

fraction of the incident radiation that is scattered into directions other than the specular direction, that produces veiling glare to degrade image contrast.

Off-Specular Peaks or ''Glints"

One further complication that can arise in connection with specular reflection concerns the occurrence of off-specular peaks or "glints" [35]. However, our approach can accomodate them

quite easily [36].

45
First, the delta-function form of the BRDF for perfectly specular reflectance can be adjusted

for such off-specular peaks. If, for example, a specular peak or glint is observed, for an unusual

reflecting surface, at dr = 6i ± a and 4>r = <f>i ± /3, that condition can be represented by a delta- function component of the BRDFfr as follows:

fr = 2-psn -Slsin^dr - smHd,±a)]-mr - {4>i±P)] [sr"'] (C14)

and, again, the value of Psp may vary with 0i,<^, and, possibly, with dr,(f)r as well, since we have

shown two possibilities for each incident direction angle with the plus-dr-minus signs, so, for complete generality, we probably should write Psp = Psp{Qi,4>i\Qr-,4>r) in eq (C14).

A similar adjustment of the expression in McCamy's notation in eq (C13) is quite straightforward.

For real reflecting surfaces, since no limitations have been placed on the form of the BRDF

/r(^M</>i;^r,</>r), it is obviously possible to express it in any way that will serve a useful purpose.

Measurement results can be just a table of values at appropriate parameter intervals, with,

possibly, smaller intervals in the vicinity of sharp peaks or other rapid changes in value, to

facilitate interpolation where required. For some purposes it may be useful to approximate these

values by a curve for an analytic function, if one can be found to serve. Such a function may turn

out to be the sum of two or more functions, one function for a continuous diffuse distribution and

one or more functions (possibly, delta functions, if the peaks are sharp enough) for the specular

peak(s) or glint(s). There is no limit to the possibilities.

IX. Appendix D. Details for Derivation of McCamy's Notation

Quite frequently, an element of reflecting surface dA is located at the focus of circular optics, so that the beam is a right circular cone with its vertex at dA. This may be true for either, or both, the incident and/or the reflected beam, as depicted in figure 4. The solid angle formed by

and containing such a beam can be specified completely by giving the angular-direction coordinates

(^Oi<Ao) of the axis along with the half-vertex angle k, as in figure 4. However, in order to use this notation, as proposed by McCamy [13], we need to be able to write the expressions for biconical

reflectance pidoi^^oiiKil^or',4>or,Kr) and for biconical reflectance factor (^o, ,</>{),,K,;^or,<Aor,'<r) in terms of the BRDF/r(0, ,(/)i ;0r,</>r) corresponding, respectively, to eqs (16) and (18). Consider a right circular cone, of half- vertex angle k , with its vertex at the origin O and its axis tilted in the direction (^o»</'o) as in figure 12. A point Pq on its axis then has the coordinates"

(Po»^o^<^'o) in the fixed system of spherical coordinates with the X—Y plane tangent to the surface

element dA at 0 and the Z-axis along the normal to dA. A plane through Pq perpendicular to the

axis of the cone, cuts the cone in a circle of radius r — po tanK. A point P on this circle has the coordinates {p,d,4>), as depicted in figure 12. Consider, also, a second, tilted set of spherical coordinates, with the same origin 0, and with

the new polar axis (the Z'-axis) along the axis of the cone. The new polar angle 9' is the angle

from that axis and, for any point P on the surface of the cone, 9' = k. The A''-axis may be

arbitrarily chosen—e.g., to lie as shown along the line of intersection between \heX—Y plane and

the tOted X' —Y' plane. Then, in the new tilted (primed) spherical coordinates, the point P on the

circle bounding the base of the right circular cone of half-vertex angle k, has the coordinates

(p',e',0') = (p,K,(/)'). (See fig. 12.) The sohd angle a> enclosed by the cone at its vertex O is easily evaluated by eq (A2-10), p. 70,

of [2] since 9h = k:

oj = 27r(l - cosk) = 47r-sin2(K/2) [sr]. (Dl)

This use of p as a spherical coordinate is confined just to appendix D. Since, when so used, fj is always explicitly designated as a spherical coordinate or

is enclosed in parentheses along with one or both of the other spherical coordinates (^,<f)), there should be no confusion with the use of p for reflectance.

46
Z"

p sin 0

Figure 12. Construction for deriving relationships between spherical coordinates and McCamys notation for right<ircular

cones.

(See figure 13 for additional details.)

47
r

• sin 0 • sin

r

• sin 0 • cos

(a) B

(b) B F (d)

D D

Figure 13. Details of construction offigure 12.

However, the projected solid angle O is not quite so readily obtained by a similar approach, except

in the special situation where the axis of symmetry coincides with the normal to the surface

element dA, so that = 0 and 6' = 6, i.e., only when the cone is not tilted. However, fl can be

obtained quite directly in all cases by the application of Wiener's construction (see fig. A2-9, p. 76, of [2]). In figure 12, if we set the spherical radius p = OP equal to unity, then the solid angle a> [eq

(Dl)] is measured by the area on the unit-radius sphere enclosed by the circle ADFCP where it intersects the cone. The radius of that circle is clearly r = p-sinK in general, and in this case, for p = 1, it is just r — sinK. Then, by Wiener's construction, the projected solid angle H =

/

cosO-d(D (note that this is cosO not cosd' or cosk, so we cannot use eq A2-14, p 74, of [2]) is given by the area of the projection of this intercepted portion of the sphere onto the plane

48
containing the element dA, i.e., by the area of the projection of the circle ADFCP onto that plane

(TheZ—K plane). It is readily seen in figure 12 that this will be the same as the area of the ellipse

BDECP', the projection of the circle ADFCP onto a horizontal plane parallel to the X — Y plane.

The area of the ellipse is clearly just cos^o times the area of the circle, which, in turn, is given by

77 • so that we can write

O. ^TT cosOo sin^K [sr] (D2)

for the projected solid angle subtended at dA by a right circular cone of half-vertex-angle k with

its vertex at dA and its axis tilted at an angle from the normal to dA

.

With these preliminary statements for (o and H, we are now ready to attack the main problem

of expressing and evaluating biconical reflectance in McCamy's notation. We start with the

general relation

p{cu,;Wr) = (l/n,)-

I fr{di,(t)i;dr,4>r)-d^r'd0.i [dimeusiouless]. (16)

First, from eq (D2), we can write fl, = tt • cos^o, - sin^K,. Next, if the usual order of integration

is reversed, the extreme limits for the full range of 0 (both 6, and 6r) are clearly d(^+K and Oq—k.

Finally, the limits of integration for </) (both and </)r) will be a function of (the corresponding) 6. In each case, the first integration, with respect to (/>, will be along lines of constant 6 (parallels of

"latitude," parallel to the X — Y plane) between the limits where they cut the circle ADFCP (see

fig. 12). These limts, then, are clearly symmetrical about the value (/)o, so that they can be written as (l>Q+a[d) and (j)Q—a(d). Before deriving an expression for the function a{6), we can now

summarize the foregoing by writing the biconical reflectance, in terms of McCamy's notation and

the function oc(d), as

P{0Qi,<l>Oi,Ki;dor,(l>Or,Kr) = (TT ' sin^K; • COS ^oi

'

fr{eu4>i-A,<^r)-

d4>r- cosQr sin^r •ddr d<^i -cosdi -sin^i ddi. (20)

In order to evaluate the function oi{d), we need to determine the equation of the circle ADFCP (fig. 12). This task can be simplified considerably by eliminating (i>o. We set (^)o" = 0, which is equivalent to rotating the fixed X—Y axes about the Z-axis to bring the A'-axis into the same "vertical" plane as the axis of the tilted cone (so that theZ—y axis now lies in the "vertical"

X—Z plane). Quantities given in the new rotated coordinates will be distinguished by a double

prime; thus, the rotated coordinates of the point Pq on the axis of the cone are

Po" = Po; ^o" = eo; <Ao" = 0; etc.

In general, the equation of a plane through a point Pq and perpendicular to the line OPq [==Po = {x^ ¥ + Zo^)*]» with direction cosines

I - xjpo, m = yo/po, « = Zo/Po

49
is

Ix + my + nz = Po = P • cos fB, (D3)

where^ is the angle between the line OPq and the line OP, and P is any point {x,y,z) in the plane. Ifj8 = K and the value of p are both fixed, we have the equation of the circle ADFCP (fig. 12) as

l"x" + n"z" = p-cosK, x" = p-sinr-cos(/)", z" = p-cosd", (D4)

because, with c/)o" = 0, jo " — Po "'sin^o "sinc^o" = 0 so that m" = 0. Then /" = sin^o " and n" =

cos do", and the values of c/)" for points on the circle that satisfy these relations are the desired

expressions for the function a (6 ). From eq (D4)

/"•sine"-cos(/)" = cosK - n"-cosr (D5)

and

cos(/)" = (cosK - ^''-cosrycr-sinr)

= (cosK - coseo"-cosO(sineo"-sine") (D6)

hence the desired limits are, recalling that 6" = 6,

a (6) = </)" = cos"' [(cos K - cos0o"cos^)/(sin^o'sin0)]. (21)

When this expression for a (6) is inserted in eq (20), we have the desired expression for biconical

reflectance in terms of the BRDF, all in McCamy's notation for right-circular-conical beams.

In the same way we can obtain, from eq (18), the expression for a biconical reflectance factor, as in eq (22).

X. Acknowledgments

The authors wish to express their sincere thanks for their support from a variety of sources

since this effort started in 1967: Infrared Information and Analysis Center'^ under ONR contract and Target Signature Analysis Center under USAF Avionics Lab. contract (Limperis, Ginsberg,

and Nicodemus); Sylvania Electronics Systems Western Division under U.S. Army Electronics

Command contract, Cintra, and U.S. Naval Weapons Center (Nicodemus); National Bureau of

Standards (Nicodemus, Richmond and Hsia). In all of this time, portions of this material have been presented or circulated, with resulting

comments and criticisms from a great many helpful individuals to whom we are very much

indebted. Our efforts were first announced and discussed at the meeting of the Optical Society of America, Technical Group on Radiometry and Photometry, in Detroit, Michigan, in October 1967. Acknowledgment is due to Applied Optics for pubhshing the letter to the editor by Nicodemus [9] to clarify some urgent matters when it was apparent in 1970 that the full paper would still be

substantially delayed. He also informally circulated rather widely a summary of much of this material in a draft intended as part of a Cintra report (that was never published). More recently, in January 1973, he reported on our efforts to the IRIS Specialty Group on Infrared Standards, at Huntsville, Alabama. We are deeply grateful for all of the suggestions, criticisms, and comments

Willow Run Laboratories, University of Michigan (now Environmental Research Institute of Michigan).

51)
that have resulted from these and other individual exchanges and we regret our inability to make

individual acknowledgments in so very many instances.

We are particularly grateful to Dr. G. J. Zissis and his staff at the IRIA Center, Environmental

Research Institute of Michigan (ERIM), for making available the small figures used to illustrate table 3. And finally, we acknowledge the very extensive review and criticism by W. H. Venable,

who made substantial contributions to our efforts.

XI. References

[1] The term "propagance" for this useful concept, the ratio of received radiance at a receiver to the emitted radiance

along the same ray at the source, was suggested by C. S. McCamy [13]. It is also defined and discussed in [2]; see

esp. eqs (2.37) and (2.38) and accompanying discussion.

[2] Part I. Concepts, Chapters 1-3, of [28], Nat. Bur. Stand. (U.S.), Tech. Note 910-1, 93 pages (March 1976).

[3] The terms "attenuance" and "scatterance" for these useful ratio concepts, which we prefer to define, like propagance,

in terms of single rays or elements of throughput by taking the ratios of radiances, are widely used by

oceanographers: see N. G. Jerlov, Optical Oceanography (Elsevier Publishing Co., New York, 1968), esp. pp. 6 & 7. [4] ANSI Standard Nomenclature and Definitions for Illuminating Engineering, ANSI Z7.1-l%7 (Revision of Z7.1-1942), RP-16, approved August 16, 1967, by American National Standards Institute, sponsored and published by

Dluminating Engineering Society, 345 East 47th Street, New York, N.Y. 10017. See also Appendix 1 of [2]. [5] Nicodemus, Fred E., Radiometry, Chapter 8 in Volume IV of Applied Optics and Optical Engineering, R. Kingslake, Ed. (Academic Press, New York, 1967): also included in [37]. [6] Planck, Max, The Theory of Heat Radiation, trans, by Morton Masius (Dover Publications, Inc., New York, 1959), Part I, Chapter I, pp. 1-21.

[7] Jones, R. Clark, Appendix [on Radiometry] in: 1. F. Spiro, R. Clark Jones, and D. Q. Wark, Atmospheric

Transmission: Concepts, Symbols, Units and Nomenclature, Infrared Physics, 5, No. 1, March 1965, pp. 11-

36. (With the help of Dr. Jones, a number of unfortunate typographical errors were corrected for its reproduction in

[37].)

[8] Venable, William H., Jr., and Hsia, Jack J., Describing Spectrophotometric Measurements, Nat. Bur. Stand. (U.S.), Tech. Note 594-9, November 1974.

[9] Nicodemus, Fred E., Reflectance Nomenclature and Directional Reflectance and Emissivity, Appl. Optics 9, No. 6, June 1970, pp. 1474-1475: also included in [37]. [10] Nicodemus, Fred E., Directional Reflectance and Emissivity of an Opaque Surface, Appl. Optics 4, No. 7, July 1965,

pp. 767-773: Erratum, .\ppl. Optics 5, No. 5, May 1966, p. 715.

[11] Chandrasekhar, S., Radiative Transfer (Dover Publications, Inc., New York, 1960). [12] Judd, Deane B., Terms, Definitions, and Symbols in Reflectometry, J. Opt. Soc. Am. 57, No. 4, April 1967, pp. 445-

452.

[13] McCamy, C. S., Concepts, Terminology, and Notation for Optical Modulation, Photographic Science and Engineering

10, No. 6, November-December 1966, pp. 314-325.

[14] International Lighting Vocabulary—3rd Edition (common to the CIE and lEC), International Commission on Illumination (Commission Internationale de I'Eclairage—CIE), Bureau Central de la CIE, 4 Av. du Recteur Poincare, 75-Paris

16', France: Publication CIE No. 17 (E-1.1) 1970. (The lEC is the International Electrotechnical Commission.)

[15] Tentative Method of Test for Specular Gloss, Federation of Societies for Paint Technology Standard No. Le-5-64; ASTM Designation: D523-62T, Issued 1953, Revised 1962, American Society for Testing and Materials (1965 Book of ASTM Standards, Part 21, pp. 109-112).

[16] Hammond, Harry K., Ill, and Nimeroff, Isadore, Measurement of Sixty-Degree Specular Gloss, J. Res. Nat. Bur. Stand. (U.S.), 44, June 1950, pp. 585-598: Research Paper RP2105.

[17] Nimeroff, Isadore, Analysis of Goniophotometric Reflection Curves, J. Res. Nat. Bur. Stand. (U.S.), 48, No. 6, June

1952, pp. 441^J48; Research Paper RP2335; also a less extended treatment in J. Opt. Soc. Am. 42, No. 8, August

1952, pp. 579-583.

[18] Kneissl, Gerhart J., and Richmond, Joseph C, A Laser-Source Integrating Sphere Reflectometer, Nat. Bur. Stand.

(U.S.), Tech. Note 439 (February 1968).

[19] DeVos, J. C, Evaluation of the Quality of a Blackbody, Physica, 20, No. 10 (October 1954) pp. 669-689.

[20] Trowbridge T. S., and Reitz, K. P., Average irregularity representation of a rough surface for ray reflection, J. Opt.

Soc. Am. 69, No. 5 (May 1975) pp. 531-536.

[21] Bracewell, Ron, The Fourier Transform and Its Applications (McGraw-Hill Book Co., New York, 1965) Chapter 5, The

Impulse Symbol.

[22] Kubelka, Paul, New Contributions to the Optics of Intensely Light-Scattering Materials. Part I, J. Opt. Soc. Am. 38,

No. 5, (May 1948) pp. 448-457; ERRATA, 38, No. 12 (Dec. 1948) p. 1067.

51
[23] Hamaker, H. C, Radiation and Heat Conduction in Light-Scattering Material, Philips Res. Rep. 2 (1947): I. Reflection and Transmission, pp. 55-67; II. General Equations Including Heat Conduction, pp. 103-111; III. Application of the Theory, pp. 112-125; IV. Various Extensions and a Generalized Theory, pp. 420-425.

[24] Richmond, Joseph C, Thermal Radiation Properties of Ceramic Materials, Proc. Symposium on Mechanical and Thermal Properties of Ceramics, Gaithersburg, Md., April 1-2, 1968, Nat. Bur. Stand. (U.S.), Spec. Pub. 303 (May

1969) pp. 125-137; also in Nat. Bur. Stand. (U.S.) Spec. Pub. 3(X), Precision Measurement and Calibration, 7, Radiometry and Photometry, H. K. Hammond, III, and H. L. Mason, Eds. (November 1971) pp. 223-235.

[25] Nelson, Richard W., On the Significance of a Simphfied Theory of Radiative Transfer, J. Opt. Soc. Am. 62, No. 11 (November 1972) p. 1368, Abstract ThC17 (paper presented at 1972 Annual Meeting of Optical Society of America, San Francisco, Cabf., October 1972). [26] Hsia, Jack J., Optical Radiation Measurements.—The Translucent Blurring Effect—Method of Evaluation and

Estimation, Nat. Bur. Stand. (U.S.), Tech. Note 594-12 (October 1976). [27] Hsia, Jack J., and Richmond, Joseph C, Bidirectional Reflectometry, Part I. A high resolution laser bidirectional reflectometer with results on several optical coatings, J. Res. Nat. Bur. Stand. (U.S.), 80A (Phys. & Chem.), No. 2, (March-April 1976) pp. 189-205; Part II. Bibliography on Scattering by Reflection from Surfaces, ibid., pp. 207-220.

[28] Self-Study Manual on Optical Radiation Measurements, Fred E. Nicodemus, Ed., in preparation by the Optical Radiation Section, National Bureau of Standards (U.S.). Chapters are being published first, as completed, as NBS

Technical Notes, Series 910-.

[29] Shurcliff, William A., Polarized Light (Harvard University Press, Cambridge, Mass., 1962).

[30] Edwards, D. K., and Bevans, J. T., Effect of Polarization on Spacecraft Radiation Heat Transfer, AlAA J. 3, No. 7

(July 1965) pp. 1323-1329.

[31] Sandus, Oscar, A Review of Emission Polarization, Appl. Optics 4, No. 12 (December 1965) pp. 1634-1642.

[32] Jones, R. Clark, Immersed Radiation Detectors, Appl. Optics 1, No. 5 (September 1962) pp. 607-613.

[33] Gershun, A., The Light Field, Moscow, 1936 (eng. Transl. by Moon and Timoshenko, J. Math. Physics, 18, May 1939,

pp. 51-151).

[34] Moon, Parry, The Scientific Basis of Illuminating Engineering (Dover, New York, 1957). [35] Torrance, K. E., Sparrow, E. M., and Birkebak, R. C, Polarization, Directional Distribution, and Off-Specular Peak Phenomena in Light Reflected from Roughened Surfaces, J. Opt. Soc. Am. 56, No. 7 (July 1966) pp. 916-925.

[36] Nicodemus, Fred E., Comment on 'Current definitions of reflectance', J. Opt. Soc. Am. 66, No. 3 (March 1976) pp. 283-285.

[37] Nicodemus, Fred E., Ed., Radiometry—Selected Reprints, American Association of Physics Teachers (AAPT), Stony

Brook, New York, July 1971, (Includes reprints of [5], [7], and [9]. [38] Page, Chester H., and Vigoureux, Paul, Eds., The International System of Units (SI), Nat. Bur. Stand. (U.S.), Spec.

Publ. 330, 1974 edition, 49 pages (July 1974).

6 U.S. GOVERNMENT PRINTING OFFICE: 1977 0—236-942

52
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U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

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1. PUBLICATION OR REPORT NO.

NBS MN -160

2. Gov't Accession

No.

3. Recipient's Accession N(

ITLE AND SUBTITLE

[GEOMETRICAL CONSIDERATIONS AND NOMENCLATURE FOR REFLECTANCE

5. Publication Date October 1977

6. Performing Organization Code

AUTHOR(S) F.E. Nicodemus, J.C. Richmond, J.J. Hsia,

I.W. Ginsberg^ and T. Limperis^^

8. Performing Organ. Report No.

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NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

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I* EG&G, Inc., Las Vegas, Nevada 89101

i^^Agro Sciences, Inc., Ann Arbor, Mchigan 48103

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

bibliography or literature survey, mention it here.) A unified approach to the specification of reflectance, in terms of both incident- and reflected- beam geometry, is presented. Nomenclature to facilitate this approach is proposed. Under specified conditions—including uniform irradiance, a uniform, isotropic, plane surface, and

allowance for edge effects due to sub-surface scattering—the geometrical reflecting properties of a

reflecting surface are readily characterized or specified in terms of the bidirectional reflectancedistribution function (BRDF). The BRDF is denoted symbolically as

fAe,A:(ir,(i>r) ^ dLA»„<t>,;er,'}>r;E,ydE,i0„<}>,) br*]

where

0 and (f> together indicate a direction, the subscript i indicates quantities associated with incident

radiant flux, the subscript r indicates Quantities associated with reflected radiant flux, Ei is incident

irradiance, Lr is reflected radiance, ana d indicates a differential quantity.

The BRDF is a derivative, a distribution function, relating the irradiance incident from one given

direction to its contribution to the reflected radiance in another direction. Nomenclature (concepts, terms, symbols, and units) for categorizing and specifying reflectance quantities for a variety of

different beam conflgurations (both incident and reflected beams) is described, and all are defined and

interrelated in terms of the BRDF. The conditions under which the formalism can be applied, including

situations involving considerable sub-surface scattering, are carefully established. The entire treatment

is limited to the domain of classical geometrical-optics radiometry and does not take into account

interference and diffraction phenomena, such as are frequently encountered with highly coherent

radiant flux. The other radiation parameters such as wavelength, (temporal) modulation, and polarization and the effects of fluorescence (or phosphorescence) are discussed briefly.

Key words: Bidirectional reflectance-distribution function; diffuse reflectance; directional reflectance; nomenclature of reflectance; reflectance; reflectance factor; reflectance geometry; reflectance nomenclature; specular reflectance; sub-surface scattering; reflection; optical reflection. KEY WORDS fsix to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper

' name; separated by semicolons) Bidirectional re flectajice- distribution function; diffuse

jlreflectance; directional reflectance; nomenclat\ire of reflectance; reflectance;

jireflectance factor; reflectance geometry; reflectance nomenclature; reflection;

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