242:(energy or photons /time/area) landing on that area element will be proportional to the cosine of the angle between the illuminating source and the normal. A Lambertian scatterer will then scatter this light according to the same cosine law as a Lambertian emitter. This means that although the radiance of the surface depends on the angle from the normal to the illuminating source, it will not depend on the angle from the normal to the observer. For example, if the
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1135:{\displaystyle {\begin{aligned}F_{\text{tot}}&=\int _{0}^{2\pi }\int _{0}^{\pi /2}\cos(\theta )\,I_{\max }\,\sin(\theta )\,d\theta \,d\phi \\&=2\pi \cdot I_{\max }\int _{0}^{\pi /2}\cos(\theta )\sin(\theta )\,d\theta \\&=2\pi \cdot I_{\max }\int _{0}^{\pi /2}{\frac {\sin(2\theta )}{2}}\,d\theta \end{aligned}}}
229:
when viewed from any angle. This means, for example, that to the human eye it has the same apparent brightness. It has the same radiance because, although the emitted power from a given area element is reduced by the cosine of the emission angle, the solid angle, subtended by surface visible to the
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Example: A surface with a luminance of say 100 cd/m (= 100 nits, typical PC monitor) will, if it is a perfect
Lambert emitter, have a luminous emittance of 100π lm/m. If its area is 0.1 m (~19" monitor) then the total light emitted, or luminous flux, would thus be 31.4 lm.
427:Ω was chosen arbitrarily, for convenience we may assume without loss of generality that it coincides with the solid angle subtended by the aperture when "viewed" from the locus of the emitting area element dA. Thus the normal observer will then be recording the same
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of a point on a surface varies by direction; for a
Lambertian surface, that distribution is defined by the cosine law, with peak luminous intensity in the normal direction. Thus when the Lambertian assumption holds, we can calculate the total
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due to the increased angle at which sunlight hit the surface. The fact that it does not diminish illustrates that the moon is not a
Lambertian scatterer, and in fact tends to scatter more light into the
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viewer, is reduced by the very same amount. Because the ratio between power and solid angle is constant, radiance (power per unit solid angle per unit projected source area) stays the same.
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The emission of a
Lambertian radiator does not depend on the amount of incident radiation, but rather from radiation originating in the emitting body itself. For example, if the
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Figure 1: Emission rate (photons/s) in a normal and off-normal direction. The number of photons/sec directed into any wedge is proportional to the area of the wedge.
339:Ω, of an arbitrarily chosen size, and for a Lambertian surface, the number of photons per second emitted into each wedge is proportional to the area of the wedge.
319:
The situation for a
Lambertian surface (emitting or scattering) is illustrated in Figures 1 and 2. For conceptual clarity we will think in terms of
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742:{\displaystyle I_{0}={\frac {I\cos(\theta )\,d\Omega \,dA}{d\Omega _{0}\,\cos(\theta )\,dA_{0}}}={\frac {I\,d\Omega \,dA}{d\Omega _{0}\,dA_{0}}}}
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Figure 2 represents what an observer sees. The observer directly above the area element will be seeing the scene through an aperture of area
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were a
Lambertian radiator, one would expect to see a constant brightness across the entire solar disc. The fact that the sun exhibits
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were a
Lambertian scatterer, one would expect to see its scattered brightness appreciably diminish towards the
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1370:. Radians and steradians are, of course, dimensionless and so "rad" and "sr" are included only for clarity.
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When an area element is radiating as a result of being illuminated by an external source, the
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Description in optics of the angular dependency of the radiant intensity of a radiant surface
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Figure 2: Observed intensity (photons/(s·m·sr)) for a normal and off-normal observer;
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of the total radiated luminous flux. For
Lambertian surfaces, the same factor of
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Modern
Optical Engineering, Warren J. Smith, McGraw-Hill, p. 228, 256
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in the visible region illustrates that it is not a
Lambertian radiator. A
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to the normal will be seeing the scene through the same aperture of area
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photons per second emission derived above and will measure a radiance of
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518:{\displaystyle I_{0}={\frac {I\,d\Omega \,dA}{d\Omega _{0}\,dA_{0}}}}
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Photometria, sive de mensura et gradibus luminis, colorum et umbrae
379:. The number of photons per second emitted into the wedge at angle
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1193:{\displaystyle F_{\text{tot}}=\pi \,\mathrm {sr} \cdot I_{\max }}
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30:"Lambert's law" redirects here. For the concept in logic, see
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photons per second, and so will be measuring a radiance of
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is foreshortened and will subtend a (solid) angle of
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757:Relating peak luminous intensity and luminous flux
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213:A surface which obeys Lambert's law is said to be
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127:Learn how and when to remove this message
1274:. Similarly, the peak intensity will be
270:is an example of a Lambertian radiator.
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1310:{\displaystyle 1/(\pi \,\mathrm {sr} )}
1487:Fundamentals of Heat and Mass Transfer
159:surface or ideal diffuse radiator is
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1417:RCA Electro-Optics Handbook, p.18 ff
560:). This observer will be recording
65:adding citations to reliable sources
1339:{\displaystyle \pi \,\mathrm {sr} }
234:Lambertian scatterers and radiators
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358:= 90°. In mathematical terms, the
274:Details of equal brightness effect
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1469:Lambert, Johann Heinrich (1760).
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1436:Pedrotti & Pedrotti (1993).
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192:. The law is also known as the
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1506:Eponymous laws of physics
1263:{\displaystyle I_{\max }}
821:{\displaystyle I_{\max }}
202:Johann Heinrich Lambert
155:observed from an ideal
1485:Incropera and DeWitt,
1440:Introduction to Optics
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32:Lambert's law (logic)
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61:improve this article
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18:Lambertian emitter
1489:, 5th ed., p.710.
1475:. Eberhard Klett.
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1384:Transmittance
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117:December 2009
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78: –
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72:Find sources:
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50:This article
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1389:Reflectivity
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59:Please help
54:verification
51:
1241:unit sphere
352:solid angle
207:Photometria
204:, from his
1526:Scattering
1516:Photometry
1511:Radiometry
1500:Categories
1455:0135015456
1405:References
556: cos(
268:black body
248:terminator
240:irradiance
215:Lambertian
87:newspapers
1348:luminance
1325:π
1293:π
1272:steradian
1220:θ
1214:
1178:⋅
1166:π
1126:θ
1110:θ
1101:
1082:π
1073:∫
1059:⋅
1056:π
1040:θ
1030:θ
1024:
1015:θ
1009:
993:π
984:∫
970:⋅
967:π
951:ϕ
944:θ
934:θ
928:
907:θ
901:
885:π
876:∫
870:π
858:∫
714:Ω
698:Ω
662:θ
656:
643:Ω
627:Ω
617:θ
611:
490:Ω
474:Ω
227:luminance
1399:Sun path
1378:See also
1364:radiance
1346:relates
1239:for the
360:radiance
344:diameter
223:radiance
1142:and so
321:photons
163:to the
101:scholar
1452:
1362:, and
1203:where
333:circle
325:energy
165:cosine
141:optics
103:
96:
89:
82:
74:
108:JSTOR
94:books
1450:ISBN
565:cos(
388:cos(
244:moon
187:cos
80:news
1366:to
1358:to
1350:to
1256:max
1211:sin
1186:max
1158:tot
1098:sin
1067:max
1021:sin
1006:cos
978:max
925:sin
919:max
898:cos
845:tot
814:max
783:tot
653:cos
608:cos
383:is
327:or
260:sun
196:or
151:or
139:In
63:by
1502::
1448:.
1444:.
1354:,
801:,
770:,
575:dA
573:Ω
569:)
546:dA
535:dA
437:dA
435:Ω
413:dA
406:dA
401:.
398:dA
396:Ω
392:)
376:dA
374:Ω
301:dA
180:=
175:;
143:,
1458:.
1333:r
1330:s
1305:)
1301:r
1298:s
1290:(
1286:/
1282:1
1252:I
1223:)
1217:(
1182:I
1174:r
1171:s
1163:=
1154:F
1123:d
1117:2
1113:)
1107:2
1104:(
1090:2
1086:/
1077:0
1063:I
1053:2
1050:=
1037:d
1033:)
1027:(
1018:)
1012:(
1001:2
997:/
988:0
974:I
964:2
961:=
948:d
941:d
937:)
931:(
915:I
910:)
904:(
893:2
889:/
880:0
867:2
862:0
854:=
841:F
810:I
779:F
732:0
728:A
724:d
718:0
710:d
705:A
702:d
695:d
691:I
685:=
677:0
673:A
669:d
665:)
659:(
647:0
639:d
634:A
631:d
624:d
620:)
614:(
605:I
599:=
594:0
590:I
571:d
567:θ
563:I
558:θ
554:0
552:Ω
550:d
542:d
538:0
531:θ
508:0
504:A
500:d
494:0
486:d
481:A
478:d
471:d
467:I
461:=
456:0
452:I
433:d
430:I
425:d
421:0
419:Ω
417:d
409:0
394:d
390:θ
386:I
381:θ
372:d
369:I
364:I
356:θ
348:θ
337:d
308:d
304:0
225:/
189:θ
185:0
182:I
178:I
169:θ
130:)
124:(
119:)
115:(
105:·
98:·
91:·
84:·
57:.
34:.
20:)
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