Atmospheric transmission: Concepts, symbols, units and nomenclature

Infrared Physics, 1965, Vol. 5, pp. 11-36. Pcrgamon Pmss Ltd., Printed in Great Britain.

ATMOSPHERIC

TRANSMISSION:

CONCEPTS,

SYMBOLS,

UNITS AND NOMENCLATURE I. J. SPIRO*, R. CLARKJONE~~and D. Q. WARK: (Receiued26 JuZy1964) Ahstnwt-This report discusses concepts, symbols, units and nomenclature for describing the transmission of radiant flux through the earth’s (variably density) atmosphere.

1. INTRODUCTION -rr*n ,4~“,.9&4~.. ,I lS&P tm”om:on;rr” fi+--,-%.X,-S. nnrth’n ntmnn~hnr~ /n+ nnr7.,n+;_ IrlO uV*l‘yn”ll “1 Lllr CI~,WIIIWJI”II “1 yvVvS4thr~..nll nll”ug” tllc. C&XV “CSILI‘ u LIuSWE.~,yLIYI” \“l Ull, .-X1able density atmosphere) requires the establishment of definitions and concepts for various atmospheric properties. Before proceeding to any consideration of absorption or emission of radiation by the atmosphere, the state of the atmosphere itself must be adequately described by temperature, mixture of gases and particles, gravity, pressure, height, andshape. A “standardatmosphere” does not consider all of these elements. 1.2 Target and background The terms “target” and “background” as defined by Kelton et al.(l) are adequate to this report and are repeated below: “ ^ L^__^L 2,. au ^.. ““,GGL ,L:,,.* C^ ~,*,,,,,4 ,“bQl.SU lo,.,,,.& “I _.. :.4P..,:G,X~ man.so ,P ;“c..~s.m4 . . . LI ra1gcr 13 L” LUC;UcLGGLal, IU~IICIII~U 1\ “J-r UlrallU “I IUlsu~U techniques. A background is any distribution of pattern of radiant flm, external to the observing equipment, which may interfere with this process. “An object may thus at one time be a target and at another be part of the background, according to the intent of the observer. An electric-power generating plant is part of the backgroundif the target is a ship moving in a nearby river, but is not if the generating plant itself is the target. Likewise, all terrain features are of interest to the users of certain reconnaissance devices and hence are targets. However, information obtained by such equipment is useful background data to other investigators. In short, one man’s target is another man’s background.” ,~...--_L-_~-

i .3 ntmospneric

n,__.._____.._

rnenomena

Consider Fig. 1. Although the distances AB, AC, AD, and AF are equal, the transmittance along these paths will vary with wavelength, altitude, humidity, and particle content. The user wants to know how much power was transmitted through the atmospheric or exoatmospheric path from the target and/or background. Knowing the transmittance of the path and the radiant incidance Q(irradiance) at the aperture of the receiver, the designer can * Aerospace Corporation t Polaroid Corporation $ U.S. Weather Bureau 8 See Section 2.3. 11

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I. J. SPIRO,R. CLARKJONESand D. Q. WARK

calculate the radiant intensity of the source; conversely, knowing the transmittance of the path and the radiant intensity of the source, he can calculate the radiant incidance at the receiver. We realize that simplifying approximations are used in accounting for such items as nonlinear variations in pressure, temperature and sun angle, as the (slant path) altitude changes.

FIG. 1. Atmospheric slant path.

The pressure and temperature dependences of spectral lines normally require a further approximation resulting from the variations along the optical path. A reasonable device is to determine the reduced optical path, that is the optical path at N.T.P. which would give the same transmittance; temperature corrections for line width and population of state are also applied where necessary. Although the results are considered to be acceptable for many purposes, it should be recognized that the methods are approximate and empirical. T. . If becomes necessary in each study to state expiicitiy the elements entering into the evahiation of transmittance such as: (a) Line intensities, whether determined from laboratory data or from theory. (‘4 Line widths at N.T.P. (4 Line widths evaluated in the nonhomogeneous atmosphere. Cd)Line intensities evaluated in the nonhomogeneous atmosphere. Source of mean transmittance values. ii Method of reduction to reduced optical path. (iiT)Temperature corrections made. (h) Models of line absorption considered, such as Elsasser, Goody, Curtis-Godson, etc. Shape of spectral lines. ;; Other refinements, such as departures from local thermodynamic equilibrium, point by point calculations through the spectrum, etc. Enough must be included to leave no doubt on the part of the reader how the results were obtained. As an aid in simplifying the computation problem, we propose that, where suitable, the atmosphere be considered a series of concentric spherical shells, with the separation points between shells being determined by how many of the atmospheric parameters remain substantially constant within that shell. The Chapman(s) designations of the various layers of the atmosphere have wide accept-

Atmospheric

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concepts, symbols, units and nomenclature

13

We propose that the definitions of these layers@) be used and that each layer be subdivided into convenient thinner layers (say lo), in each of which one or more parameters is assumed constant. These thinner layers would then be called Trl, Tr2, Stl, etc. = region of normally decreasing temperature with height up to 8-10 km Troposphere in the arctic, 16-18 km in equatorial regions. = region of increasing temperature, from the troposphere to the warm Stratosphere (0°C) isothermal layer near 50 km. = region of decreasing temperature, from the isothermal layer to the Mesosphere -:..:-..,*-.-.,,L.*:, &a-..^..“+....a .4 0” PA+.. nn tm 1111111111Lu11 aL,ll”ayllGLlG rG,lllJ+zarulGI\-- noor\ 7” L, aL L” 7” Al11. Thermosphere = region of increasing temperature, above 80 to 90 km. Upper limit not defined. awe.

2. RADIOMETRIC

UNITS

AND

CONCEPTS

2.1 Introduction The nomenclature, units and descriptions of the primary physical quantities considered most important to infrared technology are given here so that workers in this field may communicate in a common language. 2.2 Radiometric units The areferred --- rm -- ~~~ units are based on the metric system. Units differing from the given units by factors which are integral powers of 10 are acceptable. For spectral distributions, wavelengths shall be stated in microns (y), or the comparable value expressed in terms of wavenumbers (cm-l); where distributions are stated in wavenumbers (F), the equivalent in microns shall be given simultaneously. 2.3 General concepts and descriptions of radiometric quantities The concepts stated herein were proposed by R. Clark Jonesc4) to apply to the fields of both photometry and radiometry by the substitution of modifiers. A new concept (sterisent) is proposed to account for the condition when the atmosphere itself is considered as a source.

FIG. 2. Scattering angles (see Table 3, Items 3.12 and 13.

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I. 3. SPIRO,R. CLARKJONESand D. Q.

WARK

Many of the terms used in these two fields use the same term with a modifier that is either luminous or radiant: for example: luminous intensity, radiant intensity. Other terms that use these modifiers are flux, flux density, emittance, exmittance, absorptance, reflectance and transmittance. Two terms that are not constructed in this way are luminance and illuminance in photometry, and radiance and irradiance in radiometry. “I propose that the geometric concept that is common to illuminance and irradiance be termed incidance. Thus illuminance may alternatively be named luminous incidance, and h-radiance may be called radiant incidance. T ,,,-..c, ?.--..,.-~ +l.,.c :, c.“l,Ul‘“l‘ ^^_I_.. ,. I..-:..n..,a . . . ..,.rl:“~n,?. La I _..__^I_ P,l”P”Wz CL..* LUOI+h.. Ll‘FigG”Umnl~ c”Uc;GPL LIIaLIJ L” IlU,llII~U~~aIIU I~LCLIOIIUG “C termed sterance. Then luminance may be named luminous sterance, and radiance may be named radiant sterance. Photometry and radiometry share a geometrical structure that is more general than both. The geometrical structure is that of a quantity that is conserved and that propagates in straight lines at constant speed.” The very names of the subjects have the prefixes phot- and radi- built into them, as do the names of the measuring instruments: photometer and radiometer. In summary, the geometric concepts that are used in photometry and radiometry are those shown in Table 1. The centimeter and micron are used as the units of length and wavelength, respectively, in Table 1, but the meter would serve as well. Tabie i aiso inciudes the proposed pair of names sterisent and spectrai sterisent that relate to a less familiar concept. Sterisent is the sterance generated per unit path length. This concept is essential to the discussion of a fluometry in a scattering or emitting medium. For example, the luminous sterance of the daytime sky is the integral along the line of sight of the luminous sterisent at the field point multiplied by the luminous transmittance of the path from the field point to the observer. The term “exmittance” (items 2 and 8), was proposed by the Comite E.l.l. of the C.I.E. at their meeting in Paris, 3-6 February 1964. This proposal was accepted by our committee in lieu of our alternate proposal to change item 13, as constituting an international agreement and in deference to the C.I.E. l

3.1 Atmospheric attenuation-apparent source radiatio0 The attenuation of radiation by the atmosphere is a highly variable function of wavelength, and of meteorological parameters (such as the concentrations of water vapor) which are themselves highly variable and not amenable to exact determination. Being a function of the wavelength, the net attenuation of a beam which is composed of a distribution of wavelengths can be estimated only when the spectral distribution of the beam is known. Thus, in order to assess the atmospheric attenuation of radiation in a spectral band from a distant source we must know the spectral distribution of the source radiation. Atmospheric spectral attenuation is so highly variable with wavelength that this statement is usually true, even for fairly narrow wavelength bands. Before we can compute the atmospheric attenuation in order to determine the value of spectral radiant sterance (radiance), N,: or of spectral radiant intensity, J,,, at a distant source, we must know the relative spectral distribution of the source. For the reasons outlined in the preceding paragraph, there are unavoidable uncertainties involved in infrared measurements of a distant source in the earth’s atmosphere. These is a lack of complete agreement on the best methods for dealing with this difficult situation. However, it is desirable to report in as much detail as possible regarding the pertinent conditions of any measurement or calibration. These include the geometry (e.g. sun angle) and

Atmospheric

transmission:

15

concepts, symbols, units and nomenclature TABLE 1

Name of Concept

Symbol 1.

P

Flux

2.

w

Exmittance

::

J”

Incidance Intensity .

5.

N

Sterance

6.

N+

Sterisent (Path function)

7. 8.

Pi WA

Spectral flux Spectral exmittance

9. 10. 11. 12. 13.

HA JA NA NA+ c

Spectral incidance Spectral intensity Spectral sterance Spectral sterisent Emittance, Emissivity

14.

a

15.

p

16.

7

Absorptance, Absorptivity Reflectance, Reflectivity Transmittance, Transmissivity

Note 1. Note 2. Note 3. Note 4.

Note 5. Note 6.

Unit

Description Rate of transfer of energy (see Appendix) Power emergent from a surface (see Appendix) Power incident upon a surface The integral over the projected area of the sterance in that d&&on. (see Appendix) Power per unit area and per unit solid angle. (see Appendix) Sterance generated per unit path length. (see Appendix) Flux per unit wavelength interval. Exmittance per unit wavelength interval Incidance per unit wavelength interval Intensity per unit wavelength interval Sterance per unit wavelength interval Sterisent per unit wavelength interval Ratio of- ii-Emitted” radiant power to the radiant power of a black body at the same temperature (see Appendix) Ratio of “absorbed” radiant power to incident radiant power. (see Appendix) Ratio of “reflected’ radiant power to incident radiant power. (see Appendix) Ratio of “transmitted” radiant power to incident radiant power. (see Appendix)

flux flux-cm-* flux-cm-~ flux-sr-1

flux-sr-lun-s (flux-sr-i-cm-s per cm) flux-/b-r flux-cms-p-1 flux-cm-s-q’ flux-sr-l-p-l flux-sr-l-cm-s-~-i flux-sr-l-m-3-p-l

dimensioniess dimensionelss dimensionless dimensionless

See Reference 1 for descriptions recommended when only radiometry is considered. The spectral concepts 7-12 have no relevance in photometry. The name emittance was aiways used for the concepts 2 and 8, but this name was aiso often used for the concept 13. Use of the term exmittance should eliminate this confusion. In concepts 13-16, the ending -ante is used to refer to the measured value for a specific sample of a material, and the ending -ivity is used to indicate a generic property of a material. e.g., a particular sample of 347 stainless steel may have a reflectance of 0.29 whereas the pure material has a reflectivity of 0.63, at the wavelength under consideration. The concepts 6 and 12 require careful definition. In particular, the sterisent is not det!ned as the gradient of the sterance. (see Appendix) Regarding concept 13, the spectral radiant emittance or emissivity 43= WJWA,BB#~S//~A. The subscript notation Q, which could be confused with &/aA is not recommended. Simii comments apply to concepts 14, 15 and 16.

Field

Modifier

Photometry Radiometry

Luminious Radiant

Unit of Flux Lumen Watt

I. J. SPIRO,R. CLARK JONESand D. Q. WARK

16

the meteorological conditions along all ray paths, including temperature, pressure (altitude), humidity and any indications of scattering particle content. There is particular need for development of techniques for dealing practically with attenuation by scattering. 3.2 Scattering Scattering out of an elementary beam requires consideration of incidence only in the forward direction, whereas scattering into a beam demands a knowledge of the radiant sterance (radiance) as a function of incidence angle relative to the beam. The scattering coefficients must therefore be adequately described; further, the scattering coefficient for particles of a given size will differ in the regions of single and multiple scattering. Thus, the elements of scattering must include : Relative size distribution (per cent of total mass) of the particles. Mass per unit cross section of the particles in the optical path. Mass scattering coefficient of particles of a given size. Type of scattering, whether Rayleigh or Mie. Degree of scattering, whether single or multiple. Scattering angle. Shape of the particles. 3.3 “Apparent values” Often, data are reported with no adjustment for atmospheric effects; this practice (see WGIRB report(i)) involves the use of the modifier “apparent” for quantities which describe the source which would produce the same radiation at the instrument aperture if no intervening atmosphere were present. For example, J’ = TJ = HP. Where J’ = apparent radiant intensity; T = atmospheric transmittance (for the particular path and spectral beam); H = radiant incidance (irradiance) at radiometer due only to source radiation; J = radiant intensity of source; S = distance from radiometer to source. All of the radiometric quantities refer only to radiation in the beam from the source, since radiation from background sources has been ignored.

TABLE 3. ATMOSPHERICUNITS

Item no. Symbol

Name

3.1

P

Pressure

3.2 3.3 3.4

T

z

Temperature Density Altitude

3.5

h,

Scale height

3.6

S

Slant range

3.1

n

Refractive index

P

Description At the altitude (or optical path) being considered Kinetic temperature Mass per unit volume Distance above local mean sea level on a radial line from the center of the earth The equivalent height of a homogeneous isothermal atmosphere Distance from one point in space to any other along a straight line (without correction for refraction)The ratio of the speed of light in the medium to that in a vacuum

Unit dyne cm-2 “K gm cm-s km km km

Atmospheric

transmission:

concepts, symbols, units and nomenclature TABLE

Item no. Symbol

3-Continued

Name

m

Air mass

The ratio of the path length of radiation through the atmosphere at any given angle 0 to the path length toward the zenith. When e G 620, m w SAC e

3.9a

u

Optical path

J-_ p(x)

UT

3.10 3.11

Unit

Description

3.8

3.9b

17

dx where p(x) is the density of ab-

sorbing material and X is the distance along the line of sight. It should be recognized that the path may be curved by refraction The optical path at NTP which has the same transmittance as the actual optical path. Over a nonhomogenous path this requires an integration involving parameters of pressure and temperature. It is useful in spectral intervals where transmittance follows an exponential law. Negative natural logarithm of transmittance. Optical depth should not be confused with optical path rn~nc,,r~ “1 nf the 1n.c “1 nf IU”I...I”II m&nt;nn tkrm,& _A II1cu.,YI” &XXV I”.,., C’““Yb.L an attenuating medium of thickness, S. at(h) is wavelength dependent and is used for attenuation caused by either absorption scattering, or both

Reduced optical path

Optical depth Gi(&

gm cm-a

gm cm-2

at(X) = - (Is and 10 are defined below). IO at(X) is the “total” attenuation factor and is related to the attenuation factors for absorptance and scattering by the subscripts a, s, and t as in the following equation [l - at

(91 = 11- aa (4111 - 441

a corresponding notation for the attenuation coefficients in item 3.llais:

3.11a

Attenuation coefficient

3.12

4

Scattering angle

3.13a*

e

Observation angle (elev)

B

0bservation angle (AZ)

W

Mixing ratio for water vapor

3.14

* Either one should be specified-not INF.-B

both.

If lo is the initial radiation and Z8is the transmitted radiation after passing through the attenuating medium of thickness, S (in cm), Z8 = IOer*s Angle between the incoming rays from the source (usually the sun) and the observation direction measured from the interface of the scattering and clear media. (See Fig. 2) Angle between observation direction and the direction of the zenith through the scatterer. (See Fig. 2) kirnuth angie through which observation plane is rotated with respect to surface of interface between scattering and clear media. (See Fig. 2) Ratio of mass of water vapor to that of dry air in the mixture

cm-’

deg

deg deg

I. .I. SPIRO, R. CLARK JONES and D. Q. WARK

18

TABLE 3-Continued

Item no. Symbol

Name

3.15 3.16

2.

Number density Mean free path

3.17

u

Relative humidity

3.18

Rayleigh scattering

3.19

Mie scattering

3.20

Scintillation

3.21 3.22

x t

Wavelength Wavenumber

3.23 3.24

LA,A;

Frequency Bandwidth

3.25

AV Ah, AS AU

Filter Bandwidth

3.25a

Ah, AC

50 % Filter Bandwith

3.26

AX or AG

Resolution

3.27 3.28 3.29 3.30 3.31

t P w A

Q

Time (Period) Micron Millimicron Angstrom Solar constant

3.32

Y

Lapse rate

Inversion

Unit

Description Number of molecules per cm3 The average distance a particle travels between successive collisions with other particles Ratio of partial pressure of water vapor at mixing ratio w to the partial pressure at saturation Diffusion at incident radiation by particles significantly smaller than the wavelength being considered Scattering due to particles equal to or larger than the wavelength considered Fluctuationof intensity or direction of transmitted light due to temporal or spatial fluctuations in the refractive index within the atmosphere

cm-3 cm

p(micron) Spatial frequency of electromagnetic radiation. The number of waves per centimeter in a vacuum (XV= 10,000) when h is stated in I- 1.. miCrons aiiu> -Yis m cm-i. If Vand h arein the same units t = l/X Number of cycles/unit time. The range of wavelengths, wavenumbers or frequencies being considered The range between two wavelengths (or wavenumbers, or frequencies) that form the vertical sides of a rectangular, the horizontal top of which is the line passing through the peak transmittance of the filter and the horizontal bottom of which is the line of zero transmittance, that has an area equal to the area under the filter transmission curve. The bandwidth of a filter between the wavelenr&hs (or ----o---\-- ~avenumbers~ or frequencies) where the filter transmittance has fallen to 50% of the peak transmittance The smallest increment which a system can differentiate Time of one cycle lo-3mm or 1O-Bm 10d3r. or 10-gm (nanometer) 10-g or lo-lOm Irradiation per unit area (radiant incidance) normal to thedirection of the sun. Q=O*1395 W crn2 outside the atmosphere at mean earth-sun distance The decrease of temperature with height this may mean environmental lapse rate or process lapse rate, depending on the content. It is distinguished from temperature gradient, which refers to the horizontal A departure from the usual increase or decrease with height of the value of an atmospheric property. This term almost always means temperature inversion

cps p, cycle cn-r or cycle set-1 p, cycle cm-l or cycle see-’

p, cycle cm-l or cvc!e set-’

p or cm-l XC c1 X W cm-2

“C lo-r-1

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concepts, symbols, units and nomenclature

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In Table 3 the pressure, or pressure-correction equation should be specified to permit evaluation of variation in altitude. Also, wherever any units other than those listed are used, the equivalent to one of the preferred units listed in these tables should be specified, e.g., if measurements are stated as Rayleigh’s, the conversion factor of R/b. 2/X x 10-1s W cm-s sr-1 should be stated (h in CL). Acknowledgements-Many members of the committee were asked to comment and correct the original drafts of this standard. The assistance of the following is gratefully acknowledged. R. 0. B. Carpenter, Geophysics Corporation of America; K. L. Coulson, General Electric; D. D. Dudley, Analytic Services, Inc.; H. G. Eldering, Baird Atomic, Inc; F. F. Hail, Sr., ITT Federal Laboratories; J. Hamilton, Aerospace Corporation; F. S. Harris, Jr., Aerospace Corporation; J. N. Howard, AFCRL; B. J. Howell, Sperry Gyroscope Company; E. W. Kutzscher, Lockheed, California Company; R. K. McDonald, Boeing Company; F. E. Nicodemus, Sylvania Electronics Systems; A. Prostak, Bendix Systems Division; W. Wolfe, Institute of Science and Technology, U. of Michigan; G. J. Zissis, IDA.

REFERENCES 1. KELTON,G., et al., “Infrared Target and Background Radiometric Measurements; Concepts, Units and Techniques”, Infrared Physics, 3, pp. 139-169 (September 1963). 2. CHAPMAN, S., “Upper Atmospheric Nomenclature”, J. Atmos. Terr. Phys., 1, pp. 121-124 (see also letters pp. 200 and 201 of same volume) (1950). 3. U.S. Standard Atmosphere (1962), U.S. Government Printing Office, Washington (1962). 4. JONES,R. CLARK, “Terminology in Photometry and Radiometry”, J. Opt. Sot. Am., 53, p. 1314 (1963). 5. Report of Meeting in Paris 3-6 February 1964, CIE Committee E.I.I. (May 1964). 6. DUNTLEY,S. Q., R. BOILEAUand R. W. PRESENDORFER, “Image Transmission by the Troposphere I”, J. Opt. Sot. Am., 47, p. 499 (1957). 7. NICODEMUS,F. E. and G. J. ZISSIS, “Methods of Radiometric Calibration,” Report No. 4613-20-R, Institute of Science and Technology, The University of Michigan, Ann Arbor (October 1962). 8. “Smithsonian Meteorological Tables”, Smithsonian Institution, Washington, 1958. 9. ALLEN, C. W., “Astrophysical Quantities”, Athlone Press, London, 1955. 10. GOODY,R. M., “The Physics of the Stratosphere”, Cambridge University Press, Cambridge, 1954. 11. “On Uniformity of International Usage”, Physics Today (June 1962), Vol. 15, No. 6, pp. 19-30. 12. Quantities and Units of Radiation and Light (second edition), SUN Commission, IUPAP, SUN 61-51 (November 1961). 13. PLASS, G. N., “Models for Spectral Band Absorption”, J. Opt. Sot. Am., 48, No. 10 (October 1958). 14. KAPLAN, L. D. “A Methodfor Calculation of Infrared Flux for Use in Numerical Models of Atmospheric Motion”, Rockefeller Institute Press (1959). 15. COULSON,K. L., “Atmospheric Radiative Heating and Cooling”, SRI No. 2994 (March 1960). 16. Glossary of Meteorobgy, American Meteorological Sot., Boston (1959). 17. PLANCK, M., The Theory of Heat Radiation, P. Blakiston’s Sons and Company, 1914; Dover Publications, 1959. 18. NICODEMUS,F. E., “Radiance”, Amer. J. Phys., 31, 368-377 (1963). APPENDIX Introduction This appendix on radiometry is written to serve the needs of those working in the fields of infrared atmospheric transmission and the study of objects via their emission and reflection of infrared radiation. The classical approach to radiometry is to start with the total radiation of black bodies, and then by a process of analysis to develop the other radiometric concepts. This approach is perhaps appropriate to infrared studies that are oriented toward sources. If, however, one’s interest is primarily in the effect of an atmosphere on the transmission of infrared radiation, the source-oriented approach is less than fully satisfactory. In this appendix, the procedure used is just opposite to the sourcsoriented approach. Here we start with the most microscopic concept (spectral radiant sterance = spectral radiance), and by a process of synthesis we develop all the other concepts: intensity, flux-density, emittance, incidance, and power. The development is divided into two rather dissimilar parts. In Part I, Pure Radiometry, where most of the basic concepts are introduced, they are introduced in an ideal situation in which all solid bodies (except sources) are at absolute zero. In particular, radiometers are at absolute zero. The medium between the bodies is a vacuum.

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I. J. SPIRO, R. CLARK JONESand D. Q. WARK

In Part II, Applied Radiometry, the idealized conditions of Part I are relaxed to permit the atmosphere between the bodies to absorb, scatter and emit, and the solid bodies including the radiometers are supposed to be at a tinite temperature. Probably the most important way that applied radiometry differs from pure radiometry, is that in applied radiometry the radiometric quantities are not stated in terms of their total values, as they would be measured by a radiometer at absolute zero, but rather in terms of their departure from the value they have in a black body radiation field of a given temperature. In applied radiometry, all radiometric quantities are referred to some reference temperature, which is often the reference temperature of the radiometer used to make the measurement. Referred to a temperature of 3OO”K, the sterance of a 300°K black body is precisely zero at every wavelength and direction, and the sterance of a 290°K black body is negative at every wavelength and direction. In the use of reference temperatures, applied radiometry differs profoundly from photometry. Persons working in photometry have the great convenience that the walls, lenses, mirrors, bathes and detectors do not emit light. But applied radiometry at wavelengths greater than roughly 2 ~1is analogous to a kind of photometry in which all the equipment (and even the air of the laboratory) is red hot. In this appendix the concept of radiant flux is considered to be elementary. It may be considered either as a flux of photons with appropriate coherence properties, or as a classical electromagnetic phenomenon; in the latter case the radiant flux per unit area is equal to the vector cross-product of E and H. If E and H are in MKS units, the radiant flux E x His in W/ms. In pure radiometry, detectors are zero-temperature, noiseless devices that give an electrical output that is proportional to the radiant flux incident on the respective surface of the detector. In pure radiometry, the only solid bodies not at zero temperature are certain bodies specifically designated as sources; these sources are defined by their shape and their temperature in degrees Kelvin, and are assumed to be perfectly black; i.e., the spectral radiant sterance (spectral radiance) of the surface is given by

where h is Planck’s constant, k is Boltzmamr’s constant, c is the speed of light, X is the wavelength, and T is the temperature of the surface in degrees Kelvin. The radiant sterance (radiance) with all wavelengths included is given by &b(T) = oT4/7r (2) where D is the Stefat-Boltzmann

constant defined by

In this appendix we use the new radiometric terms sterance and incidence proposed recently by R. Clark Jones.t4’ Throughout the report, the spectral quantities are defined with respect to a unit bandwidth of waoelength. In no way, however, is this usage intended to imply that the alternate concepts defined in terms of the wavenumber are not equally valid.

PART I. PURE RADIOMETRY In pure radiometry, we suppose that all bodies except sources are at zero temperature, that detectors are noiseless, perfectly linear devices, and that the intervening medium does not absorb, emit or scatter. In pure radiometry, one is able to concentrate on the geometrical essentials of the subject without being distracted by the less elegant features of radiometry in the actual world. A. Spectral radiant sterance (spectral radiance) The terms spectral radiant sterance and spectral radiance have identical meaning. Spectral radiance is the conventional term. Spectral radiant sterance is the basic concept on which all the other radiometric quantities will be based. Spectral radiant sterance is a qualification of the concept of radiant flux per unit area, per unit solid angle, and per unit bandwidth of wavelength or wavenumber. It shares with the spectral radiant sterisent the property of being the most “microscopic” of the radiometric concepts. All the other radiometric concepts may be considered to be integrals over wavelength, area, or solid angle of the spectral radiant sterance. Because of its importance, we shall devote substantial space to its definition, and will consider several ways of defining it, several ways of measuring it, and its production, properties, dimensions, and units.

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transmission : concepts, symbols, units and nomenclature

21

Spectral radiance is defined with respect to a given wavelength X of the radiation with respect to a given point Q in space, and with respect to a given direction C’. 1. Symmetrical definition. We take an arbitrary point in space, denoted by Q. Through this point we take an arbitrary direction C”. On either side of the point Q we consider two imaginary areas Al and Aa, separated by the distance r (Fig. 3). The maximum diameter of both areas is to be small compared with the separation r; the condition on the smallness is that the cosines of the angles between all directions that pass through the two areas and the direction C” are to be unity to the precision involved.

FIG. 3.

We now consider the radiant flux P, within the wavelength bandwidth Ah centred at h, that passes through the two areas in directions close to that of C’. (The restriction that the directions are close to that of C’ is intended to eliminate from consideration the radiant flux that passes through the two areas in directions opposite to that of C’+.) The mean spectral radiant sterance (mean spectral radiance) & at the point Q in the direction C+, with respect to the areas A1 and AZ with separation r, is now defined as r2P RA (4 Q, C’, AI AS, 4 = ___ Al A2 AA In a purely formal way, we may now define the spectral radiant sterance mu as the limit of the mean spectral radiant sterance as Ah approaches zero, and as AI A2/r2 approaches zero in such a way that AI A2 and r all approach zero.

RA(4 Q,0

=

lim EVA(A, Q, C’, AI, Aa, 4 A.X= 0, AI A2/ra = O,AI = 0, AZ = 0, r = 0

(5)

But this limit cannot be taken in practice, so that this limit is without operational significance. So long as the areas AI and Aa are imaginary we can define (but not measure) the radiant flux passing through them no matter how small the areas are, but if we make the areas actual openings in an opaque plate, diffraction will destroy the validity of the definitions given above. Nevertheless, this limit is of great conceptual and pedagogic value because it provides a radiometric concept that is defined with respect to a point and to a direction. We shall make much use of the concept of the spectral radiant sterancc (spectral radiance) as distinct from the mean spectral radiant sterance The definition of all the other radiometric quantities will be based upon this concept. Nevertheless, it is important to recognize that the concept exists only as the result of a limiting process that cannot be achieved in practice. 2. Asymmetrical definitions. The definition of the spectral radiant sterance given above involves areas AI and AZ that are placed on either side of the point Q. This arrangement is appropriate for a point Q that is located in space, at a finite distance from surfaces of solids. If, however, the point Q is at the surface of a solid, it is necessary to modify the definition by placing one of the areas (we choose AI) at the surface of the solid, so that the point Q is located in the plane of AI (Fig. 4).

FIG. 4. With this modification in the placement of AI, the definition of spectral radiant sterance is formulated exactly as in Section A-l, In another kind of asymmetrical definition, the area A2 is moved an intinite distance from AI, so that it is

22

I. J. SPIRO, R. CLARK JONESand D. Q. WARK

FIG. 5. Geometry used in sterance definitions. described in terms of a solid angle n rather than an area (Fig. 5). With this modification, radiant sterance is defined by

the mean spectral

and the spectral radiant sterance is defined as the limit: lim

NA(A Q, CT = AA=O, Al=O,R=O

& (4 Q, C’, Al, a)

As in Section A-l, this limit is without physical meaning, but it is important as a concept and for pedagogic purposes. The second asymmetric definition of the spectral radiant sterance is the one most commonly presented, but it hides the basic symmetry of the concept of sterance. 3. Measurement. The basic idea behind the measurement of the mean spectral radiant sterance is the use of two separated apertures in opaque plates, with a detector placed at the far side of the second aperture, and with an ideal bandpass reflection type filter placed over the detector, as shown in Fig. 6.

c

DETECTOR

The plates are supposed to be black. Thus, only radiation power with wavelengths directions and lateral positions such that the power passes through the two apertures and the pass band of the filter will be effective in stimulating the detector. Because, however, the apertures in the plates are physical apertures, we must consider the effect of diffraction at these apertures. For this purpose we suppose that the apertures are approximately circular in shape and consider two parallel diameters of the two apertures, as shown in Fig. 7.

FIG. 7.

Atmospheric

transmission:

concepts, symbols, units and nomenclature

23

Suppose that perfectly collimated radiation (as from a star) is incident on the first aperture from the left. The finite size of this aperture will cause the radiation that passes through the aperture to be spread by diffraction through a range of angles whose angular width in radians is given approximately by 2X/D1. (This expression is the exact angular distance between the first zeros on either side of the central maximum for a strip of width 01.) The angular size of the second aperture as seen from the fust is Da/r. In order that substantially all the diffracted radiation light from the first aperture be received by the second, it is necessary that we have Da/r > 21/Dr (8) which inequality is symmetrical in the two diameters: D1 Da > 2hr

(9)

If we suppose that a ratio of five between the two sides of this inequality is sufficient for accuracies in the region of 1 or 2 per cent, then we have D1 Dz > IOXr

Furthermore,

(10)

if the areas AI and AZ may be considered as the squares of the diameters, we have A1 A2 > 1OOP ra

w

If the aperture AI is pushed to infinity on the left, the condition becomes dlAz > lOOh

(12)

This situation may be attained in practice by placing the detector at the focus of a lens as shown in Fig. 8. The instruments shown schematically in Figs. 6 and 8 are called radiometers (radiant phluometers in Jones’ proposed terminology). If they satisfy the conditions (11) and (12), respectively, they measure the mean spectral radiant sterance. DETECTOR

7

FIG. 8. Sterance measured. To use a radiometer, one places the point E in the entrance aperture of the radiometer at the point Q, and one places the axis K’ of the radiometer parallel to the direction C’ in which one wishes to measure the mean spectral radiant sterance. One measures the electrical output of the detector, and from its responsivity one calculates the power P measured by the detector. The mean spectral radiant sterance is then calculated by use of Equation (4) or (6). Radiometers in the form shown in Fig. 8, using a lens or a mirror, are always used in practice to the exclusion of the form shown in Fig. 6. This is done because of detector noise considerations. But the form shown in Fig. 6 represents more clearly the basic nature of a radiometer; lenses or mirrors need not be used in the construction of a radiometer. Radiometers always measure the mean spectral radiant sterance. The spectral radiant sterance cannot be measured. If the inequality in Equations (11) or (12) is satisfied, but is not too strongly satisfied, the radiometer will always be considered to be an instrument that measures sterance. But if the left-hand side is made many times the right-hand side, the instrument may be considered to be a different kind of instrument. For example, if a detector is exposed to a radiation fields so that every part of its surface sees a solid angle of 275 then the detector will usually be considered a device that measures incidence, rather than mean sterance. But this is a matter of how words are used. In the remainder of this appendix, it will be supposed that radiometers are always of the form shown in Fig. 8. Thus the geometrical properties of a radiometer may be defined by giving the shape and arithmetic magnitude A of the area of the entrance pupil, and the shape and arithmetic magnitude n of the solid angle of the radiometer.

24

I. J. S~mo, R. CLARK JONES and D. Q.

WARK

4. SpectraI radiant sterance of a source. The sterance is defined in this appendix as a field quantity-that is to say, it is a quantity that can be measured at every point in a field. In the past, however, the sterance has usually been considered to be a property of a source. In pure radiometry, it is scarcely necessary to make the distinction between the sterance at a source and the sterance in the field, because of the constancy property described in Section A-6-a; along the line from the source point Q. with the direction C+, the spectral radiant sterance measured with the point Q on the line in the direction c” is independent of the position of the point Q on the line. In anticipation of Part III, however, it is desirable to have a definition of the spectral radiant intensity of a source: If the point Q (at which the spectral radiant sterance is measured) is a point at the surface of a source, and if cos (n, C) is positive, the spectral radiant sterance so measured is the spectral radiant intensity of the source at the position Q in the direction C’. 5. Production. To produce a spectral radiant sterance, one uses a source. If one wishes to produce a spectral radiance that is zero outside the positions and directions defined by two physical apertures, one simply places a source to the left of two apertures. Sources of known spectral radiant sterance are important for calibration of a radiometer, because it is usually possible to produce a known spectral radiant sterance more accurately than one can measure separately the physical dimensions of the radiometer and the responsivity of the detector. 6. Properties: (a) Constancy along every line. The spectral radiant sterance has the very important property that in a vacuum it is a constant along every straight line between the positions where the line intersects the surface of a solid. This property is established, for example, by Plancku7) (See also a recent tutorial article by Nicodemus’r*’ ). This property, however, is not shared by the mean spectral radiant sterance. Only under conditions where the spectral radiant sterance is constant over the area and solid angle of the radiometer is the mean spectral radiant sterance independent of position along a straight line. (b) Completeness. The spectral radiant sterance is the fundamental radiometric concept, in the sense that all the other concepts are obtained from the spectral sterance by integrating over one or more of the area, solid angle and wavelength. If one knows the spectral radiant sterance at every point, in every direction, and at every wavelength in a steady-state radiation field, one has the most complete description of the radiation field that one can obtain in terms of radiometry (polarization and coherence properties, for example, are outside the field of radiometry as usually defined). 7. Dimensions and units. The dimensions of spectral radiant sterance are power per unit area, per unit solid angle and per unit interval of wavelength or of wavenumber. The corresponding units are watts per square centimeter, per steradian, per micron or per reciprocal centimeter. B. Spectral radiant intensity The spectral radiant intensity is a property of a source, and is defined with respect to a given wavelength h of the radiation, a given point Q and given direction C.

FIG. 9. Geometry for spectral radiant intensity. 1. Definition. The spectral radiant intensity of a source in a given direction is defined as the integral over the projected area of the spectral radiant sterance in that direction. With reference to Fig. 9, the spectral radiant intensity Jh is defined by J,, (h, C+) = jj Nx (A, Qs, C-9 cos (n”, ‘3

dAs

(13)

S

where the integral is extended over that part of the surface of the source for which the cosine is non-negative, and where n+ is the direction of the outward died normal. 2. Measurement: (a) Close method of measurement. In the close method of measuring the spectral radiant intensity, one uses a radiometer whose entrance area A is small compared with the projected area of the source. The spectral radiant sterance is then measured at a substantial number of different positions, Q8, on the surface of the source (all in the same given direction C+), and the integral (13) is then carried out by numerical means.

Atmospheric

transmission:

concepts, symbols units and nomenclature

25

This is a straightforward method that adheres closely to the definition. It is not, however, the method that is ordinarily employed, unless the source is supposed to have a uniform spectral radiant sterance. (b) Distant method of measurement. In preparation for this method, we note first that the definition of the spectral radiant intensity given by Equation (13) can be considered to be the product of the mean spectral radiant sterance Nh,* of the source, multiplied by the projected area aa of the source. J* (X, c+) = as RA, I) (h, C-t)

(14)

where the projected area aa is defined by a8 (C+) = JJ cos (a”, C+) dA, s

(15)

In the distant method of measurement, the radiometer is moved to a point QR at a distance R from the source such that the source can be included within the solid angle of the radiometer. It is supposed that the source measured is the only source in the solid angle of the radiometer. Then the mean spectral radiant sterance flA measured by the radiometer, being the mean over the entire field of view, is related to N,$ 8 by the same ratio as the solid angle 0.

If we note that the solid angle of the source a, is equal to As/R2, and rearrange the resulting equation, we find ls,, 8 As = n,, C2R2 (17) whence, by Equation (14), Jn (4 QR, p)

= fly (A, QR, c*) nR2

(18)

Thus, we tind that the spectral radiant intensity of the source can be measured with a radiometer if it is placed at the point Qe in the given direction from the source and at sufficient distance from the source that the source can be included within the solid angle of the radiometer. To find the spectral radiant intensity under these conditions, the mean spectral radiant sterance measured by the radiometer is multiplied by the solid angle n of the radiometer and by the square of the distance R from the radiometer to the source. 3. Properties. The spectral radiant intensity of a source has the very important property that in a vacuum in any given direction, the spectral radiant intensity is independent of the distance R from the source. In particular, the value measured by the Distant Source Method is independent of the distance R at which the measurement is made. 4. Dimensions and units. The dimensions of the spectral radiant intensity are power per unit solid angle, per unit interval of wavelength or of wavenumber. The units of spectral radiant intensity are watts per steradian, per micron or per reciprocal centimeter. C. Spectral radiant flux-density, spectral radiant exmittance and spectral radiant incidence (spectral irradiance) All three of the concepts are varieties of spectral radiant flux-density, so that they are conveniently discussed together. Spectral radiant incidence and spectral irradiance are alternate terms for the same concept. Spectral irradiance is the conventional term. The spectral radiant flux-density is the only radiometric concept for which special terms are used when the radiation is emergent from a surface (exmittance) and when the radiation is incident upon a surface (incidence). 1. Special definition. We take an arbitrary point Q in space and through it we pass an arbitrarily oriented plane with a defined normal direction n+. Consider an arbitrarily shaped single connected area A in the plane that includes the point Q. Consider the radiant flux P that passes through the area A with directions C such that cos (6, C+) is positive and with wavelengths within the interval Ah centred at A. The spectral radiant flux-density Fat the point Q in the normal direction n” is defined as the limit of the ratio -_ P AAh as the interval A.h approaches

(19)

zero and as the area A and the maximum diameter of A approach zero F(X,Q,n+)=

’ -f? ~i =‘;:A = o AA X

(20)

26

I. J. SPIRO,R. CLARK JONESand D. Q. WARK

Alternatively, with the same geometrical situation, the spectral radiant flux-density can be defined as the integral of the spectral radiant sterance over the solid angle on one side of the plane F (A, Q. 6)

= Jr N (A, Q, C+) cos (n+, C-‘) dnc

where d& is the element of solid angle in the direction C’, and where the integral is extended over the 2~ range of solid angle for which cos (n’, C+) is positive. In the important special case in which the point Q is on the surface of a solid body and the normal n” of the plane is parallel to the outwardly directed normal of the solid body, the spectral radiant flux-density is called the spectral radiant exmittance and is denoted by W, (A, Q, n+). In the important special case in which the point Q is on the surface of a solid body or is at the entrance pupil of an optical system (such as a radiometer), and in which the normal n+ is parallel with the inwardly directed normal of the body, or with the inwardly directed optical axis of the optical system, the spectral radiant flux-density is called the spectral radiant incidence (spectral irradience) and is denoted by HA (4 Q, 0 2. General definition. The definition given above in Section B-l is the more rigid form of the definition in which the radiant flux included in the definition is all the power in the 2~ steradians of solid angle on one side of the reference plane. There are many practical situations, however, in which one wishes to employ a variety of spectral radiant flux-density in which only the radiant flux from a direction within a given solid angle fld is included in the definition of the spectral radiant flux-density. This more general kind of spectral radiant flux-density is defined by F (A Q, n’, %> =

n, N (A, Q, C+) cos (n”, C+) d& JJ

(22)

With this more general definition of the spectral radiant flux-density, we then redefine the spectral radiant exmittance W, and the spectral radiant incidence HA (spectral irradiance) just as in Section B. Even more general definitions are possible in which elements of the solid angle are weighted by different amounts. D. Spectral radiant flux We define P as the radiant flux under adequately defined geometrical conditions G, in the bandwidth AX centered at the wavelength h. Then the spectral radiant flux at the wavelength X under the given geometrical conditions is defined by Ph(h,G)=

lim

P

(23)

AX=O, Ah

In particular, the spectral radiant flux passing through a surface S within the solid angle R is given by

N,, (A, Q, C’) mw+=JJnd~cJJs

cos (n+, C+) dAs

provided cos (6, C+) is nowhere negative, and where Q is on the surface S. E. Non-spectral radiometric quantities In Sections A to D, six radiometric concepts were introduced, each with the modifier “spectral”. And in Part II we shall introduce the concept of spectral radiant sterisent. In this section we introduce the six corresponding non-spectral quantities. (The count would be seven, except that the non-spectral analog of spectral radiant flux is radiant flux, which concept was introduced at the beginning as an elementary concept.) We use 0 as a generic symbol to denote any one of the six concepts: Spectral Spectral Spectral Spectral Spectral Spectral

radiant radiant radiant radiant radiant radiant

sterance NA sterisent NA* intensity Jh flux-density Fh exmittance W, incidence HA

Atmospheric The corresponding

non-spectral

transmission:

27

concepts, symbols, units and nomenclature

concepts 0 are named and denoted as follows: Radiant Radiant Radiant Radiant Radiant Radiant

sterance N sterisent N* intensity J flux-density F exmittance W incidence H

1. Special definition. Given the spectral concept 0, (h), the corresponding defined by

non-spectral

concept 0 is

Non-spectral concepts defined in this way are sometimes named “total” (e.g., total radiant exmittance) to indicate that the integration is over ail wavelengths. 2. General ~~n~fio~ The definition just given in Section E-l is the more rigid form of the definition in which the non-spectral concepts in&de the radiation at all wavelengths. There are many practical situations, however, in which it is convenient to employ a variety of non-spectral concepts in which only the radiation within the band from XI to XZis included in the concept. This more general kind of non-spectral concept is defined by ;I Oh(h)

@(Xl, h2) =

dA

(26)

5

Even more general definitions are possible in which a wavelength dependent

weighting function is used.

F. S%mmary The concepts developed in this part and in Part II are summarized in Table 4, along with the corresponding symbols and units. The minimal set of other quantities on which the concept depends are listed in parentheses following the symbol. The modifier “radiant” may be omitted when no confusion with the corresponding photometric qualities can-. PART II-APPLIED RADIOMETRY In this Part II on Applied Radiometry we relax the idealizations made in Part I. In particular, (1). The medium between the solid bodies is not a vacuum, but is an atmosphere that may absorb, scatter and emit radiation. (2). The solid bodies and the radiometers are not necessarily at zero degrees Kelvin. (3). The sources are not necessarily perfectly black, but may have an emissivity different from unity. (4). Detectors are not free of noise. Furthermore, we take cognizance of the fact that sterance is not measured with a zero temperature radiometer, and introduce the concept of the reference temperature. Suitable language to distinguish between source properties and field properties is introduced in Section J. In the presence of scattering atmospheres, it may be noted that the usefulness of the concept of intensity depends on the scattering not being so great that radiation from distinct sources gets mixed up. If the observer is in the middle of a dense sunlit cIoud, he would describe the radiation field by stating the variation of sterance with direction; the intensity of the sun at a point inside the ctoud would not be a useful concept. B. Reference temperature of a r~iometer A very important radiometric concept is that of the reference temperature of a radiometer. When a radiometer is at a temperature above zero degrees Kelvin, a radiometer facing a source at zero degrees Kelvin will have a negative reading; the radiometer will radiate to the source. With every radiometer, there is some source temperature Trer such that when it fills the area and solid angle of the radiometer, the radiometer gives a zero reading. The temperature Trer is called the reference temperature of the radiometer. With some radiometers, the reference temperature is close to that of the chopper, and may vary with the ambient temperature. Some radiometers employ an internal temperature-controlled source that serves to set the reference temperature. Still other radiometers have no response to a steady radiation field, and thus measure only differences of sterance or intensity; such radiometers have no reference temperature. C. Reference temperature of a radiometric quantity When a radiometric quantity is measured with a radiometer with the reference temperature reference t~~~t~ of the ra~ome~c quantity is equal to Trer.

Trel, the

28

I. J. SPIRO, R.

CLARK JONES and

D. Q.

WARK

TABLE 4. RADIOMETRIC QUANTITIES

Name

Radiant Radiant Radiant Radiant Radiant Radiant Radiant Spectral Spectral Spectral Spectral Spectral Spectral Spectral Radiant Radiant Radiant Radiant

Flux flux-density exmittance incidence intensity sterance sterisent radiant flux radiant flux-density radiant exmittance radiant incidence radiant intensity radiant sterance radiant sterisent emittance, radiant emissivity absorptance, radiant absorptivity reflectance, radiant reflectivity transmittance, radiant transmissivity

Symbol P

F(Q,O W(Q) H(Q) JCQ, '3 NQ,'3 N*(Q,c,T) PA F(4Q,0 WAC&Q> Hx(kQ, JA(A, Q,C+> NC4 Q,'3 T;N Q,C+,T) 48 PC4 44

Unit W W-cm-2 W-cm-2 W-flux-cm-a W-flux-sterac-l W-fl~-sterad-1-cm-2 W-flux-sterad-~-cm-3 w-flux-~-’ W-fl~-cm-2-~-1 W-flux-em-2-@ W-flux-cm-+-l W-flux-sterad-l-~-l W-flux-sterad-+m-2-~-1 W-fl~-sterad-1-crn-3-~-1 dimensionless dimensionless dimensionless dimensionless

We first review the concepts of Part I under these relaxed conditions, then define a few more concepts (sterisent, absorptance, transmittance, reflectance, emittance and reference temperature) and then finally discuss the additional attributed one must assign to any statement of sterance or intensity in applied radiometry. In Part II we suppress the modifier “radiant” in the radiometric terms. The modifier is superfluous if the only phluometry under discussion is radiometry. A. Status of Part I concepts in applied radiometry All the radiometric concepts included in Part I are defined as field concepts-that is to say, they are defined as concepts that are measured at an arbitrary point in the space between the solid bodies. Accordingly, these concepts may be taken over without change in applied radiometry, with three qualifications: 1. The property that the sterance and intensity are constant along a line holds in a vacuum, but does not hold when the medium has absorption, scattering or emission. 2. Nearly always in applied radiometry, the statement of a radiometric quantity is referred to some reference temperature T,,I. 3. The intensity requires special consideration, as follows. In Part I it was not necessary to distinguish between the intensity measured at a source, and the intensity of a source. The close method of measurement of Section B-2-a of Part I yields the spectral intensity of the source at the source. The distant method of Section B-2-b yields the spectral intensity of the source at a distant field point. In a vacuum these are necessarily the same (in the same direction C+). But in a real atmosphere the spectral intensity of a source will depend on the distance from the source. Thus, although the spectral intensity is always referred to a source, it is a field concept that in a real atmosphere depends on the distance from the source. Specifically, the value of a radiometric quantity referred to the temperature T is the value of that quantity referred to zero degrees Kelvin minus the value of that quantity in a black body radiation field that has the temperature T. Thus, for example, the spectral sterance NA, T referred to the temperature T is equal to the spectral sterance NA, a, referred to absolute zero minus the spectral sterance NA, M,, referred to absolute zero of a black body radiation field with the temperature T:

Since all the intensive radiometric concepts of Part I may be considered to be defined in terms of the spectral sterance, the above formula may be used to change the reference temperature of a radiometric quantity or to reduce the quantity to the value corresponding to a reference temperature of zero degrees Kelvin.

Atmospheric

transmission:

concepts, symbols, units and nomenclature

29

D. Special sterisent The spectral sterisent and the sterisent are the only intensive radiometric concepts that need to be added to those of Part I. These concepts could not be introduced in Part I because they have a value of zero in a vacuum. The sterisent is different from zero only in an atmosphere that scatters or emits radiation. Like spectral sterance, spectral sterisent is defined with respect to a given wavelength Xof the radiation, with respect to a given point Q in the atmosphere, and with respect to a given direction C’. The sterisent is the sterance generated per unit path length of the atmosphere. 1. Definition. To define the spectral sterisent, we start out exactly as we did for spectral sterance in Part I. We consider an arbitrary point Q in space, and the direction C’ through this point. Then we set up two imaginary areas Al and AZ, separated by the distance r. The maximum diameter of each area is to be small compared with the distance r, with the same condition of smallness as in Section A-l of Part I. We now consider the radiant flux PT that is generated in the space between the two apertures such that the power is generated in a direction and position such that the line of direction passes through both apertures in a direction close to that of C+, and such that the power lies within the band AX centered on X. The power PT may be generated either by emission, or by scattering into the direction C-from some other direction. As indicated by the subscript T, the power PT is referred to the reference temperature T. The mean spectral radiant sterisent &* at the point Q in the direction C’, with respect to the areas AI and AZ with separation r, is now defined as

lis,* (4 Q, C’, Al, Aa, r, T) =

AsA

(27)

Note that the power of r in the numerator is one, not two as in the definition of spectral sterance. In a purely formal way, we may now define the spectral radiant sterisent NA* as the limit of the mean spcctral radiant sterance as Ah approaches zero, and as AI AzJr approaches zero in such a way that AI, AZ and r all approach zero. &* (4 Q, C’, T) =

R* (4 Q. C, A, Aa, r, T)

lim Ar\=O,A1A,/r=O,AIO,r=O

(28)

But this limit cannot be taken in practice, so that this limit is without operational significance. So long as the apertures A1 and AZ are imaginary we can detine (but not measure) the power generated between them no matter how small these areas are, but if we make the apertures physical apertures, diffraction will destroy the validity of the definitions given above. Nevertheless, this limit is of great conceptual and pedagogic value, becuase it provides an important radiometric concept that is defined with respect to a point and a direction. We shall make much use of the concept of spectral sterisent as distinct from the mean spectral sterisent. Nevertheless, it is important to recognize that the concept exists only as a result of a limiting process that cannot be carried out in practice. 2. iUeasurement. The basic idea behind the measurement of the mean spectral sterisent is the use of a radiometer that is focus& upon a black surface with a temperature equal to the reference temperature of the radiometer. In Fig. 10 the lens of the radiometer focusses the black surface with area AI on the detector. The lens has the area AZ.

I

I

K \ BLACK

SURFACE

FIG. 10. Measurement

i

I I u DETECTOR

v LENS

of spectral sterisent.

The image of the black surface at the detector should be slightly larger than the detector. The areas A1 and AZ and the separation r must satisfy the diffraction conditions of Section A-3. Part I, as well as the cosine condition of Section A-I of Part I. In order that the scattered radiation be substantially the same as in the absence of the radiometer, it is necessary that the solid angles Al/r2 and Al/r2 be small. Particularly is scattering at small angles is important, it is necessary that Al/r2 be small; specifically, Dl/r should be. small compared with the mean scattering angle of the atmosphere. When the space between AI and Aa is a vacuum, the output of the radiometer is exactly zero, provided

30

I. J.

SPIRO,

R.

CLARK

JONESand D. Q.

WARK

that the surface is perfectly black and that the temperature of the surface is exactly equal to the reference temperature of the radiometer. When, however, at atmosphere is admitted to the space between the lens and the black surface, the radiometer will in general provide a non-zero reading, corresponding to a power Pr incident on the detector. In order to measure the spectral radiant sterisent at the point Q in the direction C+, the radiometer shown in Fig. 10 is placed with E at the point Q, and with the optical axis K’ of the radiometer parallel to the direction C. The power PT incident on the detector is then calculated by use of the responsivity of the detector. The power PT is then substituted in Equation (27) in order to determine the measured mean spectral radiant sterisent is,*. The measurement yields the mean spectral sterisent. The spectral sterisent cannot be measured. No equipment is known to have been built for the measurement of sterisent in the infrared. For use in the visible region, however, Dr. S. Q. Duntley and his colleagues have constructed excellent equipmentczO). 3. Properties. Like the other intensive radiometric quantities, the measured spectral sterisent is referred to a reference temperature T. Unlike the other quantities, however, there is no simple way to predict how the spectral sterisent will change if the reference temperature is changed. Within the framework of radiometry, the only way to determine the spectral sterisent for another reference temperature is to measure it with another reference temperature. 4. Dimensions and units. The spectral sterisent has the dimensions of power per unit solid angle per unit volume per unit interval of wavelength or of wavenumber. The corresponding units are watts per steradian per cubic centimeter per micron or per reciprocal centimeter. E. Sterisent Sterisent is defined in terms of the spectral sterisent in the same way as the other non-spectral quantities (see Part I, Section E). The sterisent has the dimensions of power per unit solid angle per unit volume. The corresponding units are watts per steradian per cubic centimeter. F. Transmittance The transmittance, 7, of a solid body or of an atmosphere, is the ratio of the output radiant power to the input radiant power. Usually, the input radiant power is in the form of a well-colimated beam, and the output power measured is usually the power that remains with the collimated beam. Sometimes, however, all the output power is measured, including that which has been scattered through large angles. In paper technology, and in photography, it is customary to define a number of kinds of transmittance, called specular, diffuse, and double-diffuse. The transmittance usually depends on the wavelength, the geometry of the input power, and the geometrical configuration of the equipment that measured the output power. No statement of a transmittance is complete until the geometrical conditions are specified. In the special case of atmospheres, the presence of absorption by vibration and rotation lines produces a transmittance that varies rapidly with wavelength. Measuring equipment of very high resolving power is necessary to determine the transmittance for strictly monochromatic light. Because of the very rapid variation of transmittance with wavelength, it is customary to use transmittances that are averaged over some finite band of wavelengths. These average transmittances are more convenient to use than the monochromatic transmittances, but they have the disadvantage that they do not exhibit exponential absorptance (Beer’s law). The subject of the transmittance 7 (X) of atmospheres is treated at length in another part of the proposed standard. G. Reflectance The reflectance, p, of a solid body is the ratio of the reflected radiant power Atmospheres are not ordinarily considered to have a reflectance. Just as with the transmittance, the reflectance depends on the wavelength, power, and the geometrical configuration of the equipment that measures the of the reflectance is complete until the geometrical conditions are specified. larly important if the reflection is diffuse.

to the incident radiant power. the geometry of the incident incident power. No statement These conditions are particu-

H. Absorptance The absorptance a of a solid body or an atmosphere is the ratio of the absorbed power to the input power. With a solid body, the absorptance is usually taken to be the complement of the sum of the transmittance and reflectance : a(A) = 1 - T(h) - p(X) (29) With an atmosphere,

the absorptance

is the complement

of the transmittance:

a(X) = 1 - T(X)

(30)

Atmospheric

transmission:

concepts, symbols, units and nomenclature

31

The absorptance of an atmosphere is the sum of the effects of two quite different mechanisms. The first mechanism may be called true absorption and represents the absorption of a quantrum of the radiation by a molecule, and subsequent degradation to heat. Some of the power of the true absorption may be m-emitted at a different wavelength (fluorescence), but this phenomenon lies beyond the scope of this standard. The second mechanism of absorption is scattering, by which the power is changed to a direction where it is not included in the output power. The analysis of the effect of scattering may become complex if a major part of the unput radiation is scattered through large angles as it is in stellar atmospheres (see S. Chandrasechar, “Radiation Transfer”). I. Emittance emissivity The emittance, c, (alternatively called emissivity) of a solid body is the ratio of the measured emittance to the emittance of a black body at the same temperature, where both emittances must be referred to zero degrees Kelvin. The emittance usually depends on the wavelength, the angle between the surface normal n+ and the direction C’ of observation. The emittance may be the ratio averaged over part or ah of the solid angle. No statement of the emittance is complete until the geometrical conditions are specified. The concept of emittance may be generalized to apply to flames, or to a specified quantity of an atmosphere. The emittance of flames and atmospheres will, like the transmittance and absorptance, vary rapidly with the wavelength. J. Radiometric attributes of a source When a source is observed in an atmosphere that absorbs, emits or scatters, there are a number of attributes of the measurement for which it is convenient to have descriptive terms. These terms may be applied to the sterance and to the intensity of a source. Close, distant-the distant intensity of a source is that measured from a distance. The close intensity of a source is that measured close to the source. Relative, absolute-the relative sterance of a source is the sterance of the source minus the sterance (“background” sterance) read when the radiometer is translated parallel to itself to a position where the source is not in the field of view. The absolute sterance is the sterance of the source. Referred, reduced-a referred sterance is a sterance referred to a temperature above absolute zero. A reduced sterance is the sterance referred to zero degrees Kelvin. To illustrate how these terms are used in combination, we consider a distant source that is located in a part of the sky that except for the source has uniform sterance. Then as a radiometer is moved so that the source is alternately in and out of the solid angle of the radiometer, the difference between the two mean sterances can be converted into an intensity by the method described in Section B-2 of Part I. This particular intensity depends on the distance R of the source because of the absorption of the intervening atmosphere, and it depends on the sterance N of the background, as indicated by the symbol J (R, N). This particular intensity is the “distant and relative” intensity, and is the effective intensity for any scanning system that seeks to detect the source at the distance R with the given background. By straightforward calculation it is possible to find the intensity without reference to the background. When this is done, the intensity depends on the reference T of the radiometer that measures the background sterance N. This intensity is denoted J(R. T) and is called the “distant. absolute and referred” intensitv. If one knows enough agout the properties of the atmosphere, it is possible to determine the intensities J(R, N) and J(R, T) when R is very small or zero. Alternatively, the intensities close to the source can be measured with a radiometer close to the source. These intensities no longer depend on the distance R. The intensity J(R, N) becomes J(N) and is called the “close and relative” intensity. Similarly, the intensity J(R, 2’) becomes J(T) and is called the “close, absolute and referred” intensity. The second intensity J(R, T) may have its reference temperature changed to absolute zero. This intensity is denoted J(R) and is called the “distant, absolute and reduced” intensity. Finally, the fourth intensity J(T) may have its reference temperature changed to absolute zero. This intensity is denoted J and is called the “close, absolute and reduced” intensity. These symbols and terms are summarized in Table 5. TABLE5. Distant and relative Distant, absolute and referred Close and relative Close, absolute and referred Distant, absolute and reduced Close, absolute and reduced

J(R, N) JR 7’) J(N) J(T) JO J

32

I. J. SPIRO, R. CLARKJONESand D. Q. WARK

K. An iIlustrative problem In this section we present and solve an illustrative problem that involves the calculation of the radiant ntensity of a source from measurements made at a distance with a radiometer. 1. Statement of the situation: (a) Radiometer. The radiometer is equally responsive to sterance of all wavelengths, and is calibrated in mean sterance. The reference temperature of the radiometer is 3OO”K, and the field of view is a square 0573 deg = @Ol radian on a side. The solid angle n of the radiometer is thus 1O-4 steradian. (b) Source.The source to be characterized is a spacecraft at a distance of 10 km that presents to the radiometer a rectangle 20 m by 50 m. Since the field of view of the radiometer at 10 km is 100 m by 100 m, the spacecraft can be included within the solid angle of the radiometer. The spacecraft is perfectly black and has a temperature of 295°K. The spacecraft is viewed along a slant path (see Fig. 11).

FIG. 11. Geometry for problem.

(c) Atmosphere. A much oversimplilied model of the atmosphere will be used. The atmosphere does not scatter, it is at a uniform temperature of 290”K, and its absorptance is the same at all wavelengths. The transmittance from the spacecraft to the radiometer is 60 per cent, and the transmittance from the radiometer along the slant path to outer space is 5 per cent. The sterance of outer space is zero referred to absolute zero. 2. Calculation of the sterances. In this section we calculate the sterances that the radiometer will mesaure. We shall need to use the sterance S(T) of a black body at the temperature Treferred to absolute zero, when T has the values 300,295,290, and 0 degrees: S(300) S(295) S(290) S(0)

= = = =

14,500~ W/cm2-ster 13,550~ W/cm2-ster 12,650~ W/cm2-ster 0

(31)

First, we calculate the sterance along a line that does not intersect the spacecraft. The sterance along this line referred to absolute zero is 95 per cent of the sterance S(295) plus 5 per cent of the sterance S(0): N(R) = 0.95 S(290) + 0.05 S(0) = 11,970p W/cm2-ster

(32)

(Because the absorptance of the atmosphere is 95 per cent and because it has the uniform temperature of 290 degrees, it behaves like a black body with an emittance of 95 per cent.) Second, we calculate the sterance along a line that intersects the spacecraft. The sterance along this line is 60 per cent of the sterance S(295) plus 40 per cent of the sterance S(290): Na(R) = 0.6 S(295) + 0.4 S(290) 13,190 pW/cm2-ster As indicated by the notation, both of these sterances are distant absolute, and reduced sterances.

(33)

Atmospheric transmission: concepts, symbols, units and nomenclature

33

Next, we change the reference temperature from absolute zero to the reference temperature of the radiometer, which is 300°K. This is carried out by substrating S(300) from each of the last two sterancesi We find N(R, T) = N(R) - S(300), T = 300 deg

= -2530 pW/cma - ster

(34)

and NA(R, T) = NA(R) - S(300), T = 300 deg

= - 1310ccW/cm*-ster

(35)

These sterances are distant, absolute and referred sterances. They are negative because the radiometer is warmer than the source and the atmosphere. Finally, we calculate the mean stemnces measured by the radiometer. When the spacecraft is not in the field of view, the sterance is uniform over the solid angle of the radiometer, so that l&R, T) = N(R, T) = - 253OpW/cm~ -ster,

T = 300 deg

(36)

When the spacecraft is in the field of view of the radiometer, the spacecraft covers 10 per cent of the field of view. The mean sterance therefore is 10 per cent of the sterance NA(R, T) plus 90 per cent of the sterance N(R 7% RA(R, T) = 0.1 NA(R, T) = 0.9 N(R, T), T = 300 deg

E - 241OpW/cms - ster

(37)

The last two sterances are. distant, absolute, referred and mean sterances (they are also radiant sterances). 3. Statement of theprobiem.We are now ready to state the problem. Given the two mean sterances of Section K-2, and the information about the radiometer and about the atmosphere in Section K-l (but not given the information about the source). the task is to calculate the several radiant intensities of the source. 4. Solution of the problem. The distance R and presented area of the spacecraft must be known. It is supposed that these can be obtained by visual observation. We recall from Part I, section B, that the general procedure to calculate the radiant intensity from a mean sterance, is to multiply the mean sterance by the solid angle n of the radiometer and by the square is the distance R. The RRa product has the value !ARa= lo-4 (10”)s = 108cm2

(38)

First, we calculate the distant and incremental intensity: J(R, N) = (NA(R, T) - m(R, T)) RR2 = (120rW/cma-ster)

10s cm2

= 12,000 W/ster

(39)

Next, we calculate the distant, absolute and referred intensity. To do this, we must first determine the (unbarred) sterance NA(R, T) from the radiometer measurements. We do this essentially by using Equation (37) backward. If we solve Equation (37) for NA(R, T), and use the fact indicated by Equation (36) that the barred and unbarred N(R, T)‘s are equal, we tind NA(R, T, = 10 fl~(R, T) - 9 fl(R, T), T = 300 deg

= 10 (- 2410) - 9 (- 1,310) = - 131O~W/cn+ - ster

(40)

This is the distant, absolute and referred sterance, with the reference temperature 300 degrees. More precisely, since we are not permitted to know that the source has a uniform sterance, it is the mean sterance, where by mean we imply it is averaged over the projected area of the source. rNF*-c

34

I. J. SPIRO,R. CWRK Jonas and D. Q. WARK

To Fordthe corresponding radiant intensity, we must now multiply NA(R, T) by the solid angle n, of the source and by the square of the distance J(R, T) = NA(R, T)il,RsT = 300 deg

= (- 131O~W/cm*--ster) 10’ cm* =-

13,100 W/ster

(41)

This is the distant, absolute and referred radiant intensity of the source, referred to 300 degrees. We shah shortly wish to shift the reference temperature ofthe intensity. To do this we must know the intensity Z(7) of a black body source with the termperature T with the same area (1000 ms) as that of the source, and referred to absolute zero. These are Z(300) = 145,000 w/ster Z(295) = 135,000 W/ster Z(290) = 126,500 W/ster Z(0) = 0 We have now cakulated two of the three distant intensities. The third is the distant absolute and reduced J(R), to and calculate J(R) we must add Z(300)to the intensity J(R,SOO) given by Equation (41) J(R) = J(R, T) + Z(300), T = 300 deg

= - 25,300 + 145,009 = 119,700W/ster

(43)

It remains to calcuiate the three close intensities. It eliminates a lot of complexity if we note that the sterisent of the atmosphere is zero if we refer the radiometric quantities to the temperature 290 degrees of the atmosphere. With this reference temperature we need consider only the absorption of the atmosphere; the emission is zero. The distant absolute and referred intensity J(R, T) referred to 290°K is J(R, T) = J(R) - Z(290 deg), T = 290 deg

= + 5400 W/ster

(44)

Because the transmittance of the atmosphere is 60 per cent, this intensity is 60 per cent of the close, absolute and referred intensity J(T) J(T) = J(R, T)/O*6, T = 290 deg

= 9OtMW/ster

(45)

With the reference temperature 3OO”K,this is J(T) = 9ooo + Z(290) - Z(300)

= - 9500 WJster

(46)

With the reference temperature of absolute zero, we have the close absolute and reduced intensity J J = J(T) - Z(300), T = 300 deg

= - 9500 + 145,Ooo = 135,500W/ster which value of J is the intensity of a 295 degree spacecraft referred to absolute zero.

(47)

Atmospheric transmission: concepts, symbols, units and nomenclature

3s

It remains now only to calculate the close and relative intensity J(N). J(N) is the difference between J(T) and the intensity of 1000 ma of background. We choose to caiculate the sterance of the background with reference to 290 degrees, because the emission of the atmosphere is zero with this reference. Because the transmittance from infinity to the radiometer is 5 per cent, and the temperature from the spacecraft to the radiometer is 60 per cent, it follows that the transmittance from infinity to the spacecraft is 8.333 per cent. The sterance of the background referred to 290 degrees is therefore 8.333 per cent of the sterance of a black body at absolute zero : Na = - 0.09333 S(290) = - 1052r*wlcms-ster

(48)

whence the radiant intensity of 1000 ms of the background is JB = - 10,502 W/ster

(49)

The close and relative intensity is then given by J(N> = J(T) - JB, T = 290 deg

= 9000 - ( - 10,520)

(SO)

This concludes the solution of the problem. The results, with respect to the intensity, are summarized in Table 6. T.ULE 6. J(R, ZV)= 12,080 W/ster

J(R, T) J(N) J(T)

= - 13,100 W/ster, T = 300 deg

= 5400 W/&r, T = 290 deg = 19,500 W/ster = 9500 W/ster, T = 300 deg

= 9000 Wjster, T = 290 deg

JO

= 119,700W/ster

J

= 135,500 W/ster

5. Father considerations about the sterisent. Because of the very simple structure of the atmosphere in the ilhrstrative problem, it was not necessary to calculate the sterisent of the atmosphere. We here calculate the sterisent of the atmosphere. We suppose for this calculation that the atmosphere is uniform between the spacecraft and the radiometer. Then the exponential absorption coethcient k of the atmosphere is the solution of

e

-_(lO*cm)k =

1 _

0.6 =

0.4

whence k = 0.915 x lo-“/cm

(51)

The coefficient k is the absorption coethcient per centimeter and, since k is small, it is also equal to the absorptance of a one centimeter path. By Kirchhotf’s law, the emittance is equal to the absorptance. Thus the sterisent, which is equal to the sterance generated per unit path length, is equal to k times the sterance of a 290 degree black body: N* - k [Z(B) - Z(T)]

(52)

36

I. J.

SPIRO,

R.

CLARK

JONESand D. Q. WARK

where T is the referencetemperature. This result is in agreement with the fact used in Section K-4 that the sterisent is zero when the reference temperature is 290 degrees. Nowhere in this report have we yet given the general formula for the relation between sterance and sterisent. We wish to evaluate the spectral sterance N,( X, Q, C+, T) at the wavelength h at the point Q in thedirection C with the reference temperature T, in terms of the spectral sterisent along the line from Q in the direction C+, and in terms of the spectral sterance of a solid body along the line, or in terms of the spectral sterance at infinity. The spectral sterisent at the point Q’ on the line in the direction C” is denoted by NB* (X, x, C’, T) where x is the distance from Q to Q’. The transmittance from the point Q’ to Q is denoted by I (X, x). First we consider the case where the line intersects a solid body S at the point Q” at the distance x8 (see Fig. 12). Then we have NA (4 Q.

I

c+,T) = f’

7(X, x) NA*(X,x, C+, T) dx +

n


(53)

SOURCE

Frc. 12. Spectral sterisent relationships.

where the reference temperatures of both sterances and sterisent must be the same. The tirst term is the (absolute and referred) par/r spectral sterunce, and the second term is the product of the path transmittance and the close, absolute and referred spectral sterance of the solid body. Second, we consider the case where the line extends an infinite distance (example, a path through the atmosphere that does not intersect a celestial body). Then we have

N,(h Q, C’, T> = ; 7(X, x) Nx*(X, x, C+, T)dx + 7(X, 03) NA(& to, c, J

T)

Even in the special case where the spectral sterance at infinity is that of a black body at absolute zero, the last term will not be zero. A special property that has been defined by Duntleycs) is the equilibrium sterance. The equilibrium sterance at any point in the medium is the sterance one would have at that point if thesterisent and absorption coefficient have everywhere the values they have at Q. Under this hypothesis, 7(X, x) has the form ~(h, x) = e--M&

(55)

and 7(X, co) = 0

(56)

Then we have from the last three equations, N,@) (A, Q, C+, T) = Nh*(X, Q, C+, T)[r

e- Wx d.x

= NA*(A, Q,C+, T) k(A) Thus the equilibrium sterance differs from the sterisent only by a factor equal to the absorption coefficient. Accordingly, one may use the equilibrium sterance instead of the sterisent if one wishes: either one will serve the purposes of the other.