Measurement of Spectral Radiance

Spectral Radiance Formula

Symbol: Ll

Unit: W∙m-2∙sr-1∙nm-1

Where dΦ(λ) is the spectral radiant power passing through an infinitely small area enclosing that point and propagating within the solid angle, dΩ, in the given direction, to the product of the wavelength interval, dλ, and the area of a section of that beam on a plane perpendicular to this direction (dA.cosθ) containing the given point and to the solid angle, dΩ.

Spectral Radiance Diagram

Spectral radiance includes a crucial component, that of étendue, or geometrical extent, defined as:

Spectral Radiance Formula

For a beam propagating through loss-less, non-diffusing media, the quantity G.n2 , where n is the refractive index, is invariant. Spectral radiance provides information on the light coupled by an optical system, such the eye, a lens or a monochromator and is invariant and independent of distance.

Spectral radiance may be measured by two manners, using an imaging technique or indirectly through an irradiance measurement. In both cases, the measurement is performed in a specific field of view (FOV) or solid angle of acceptance (often described by a planar angle, γ), defining the solid angle area of the source measured. 

Spectral Radiance Diagram

The imaging technique employs a telescope to relay an image of the source on a measurement plane. The lens of the telescope defines the solid angle of measurement whilst an aperture at the image plane defines the area of the source measured though a circularly symmetric solid angle field of view (FOV). 

LINK: TEL301 Direct View Telescope

The selection of field of view is defined by the application. In spectroradiometry, spectral radiance is used to measure the luminance (often termed brightness) of a display such as a screen. In this case, it is crucial that the field of view is over-filled by the source. In photobiological safety testing, it is no spectral radiance, but spatially averaged radiance of a FOV that is chosen to replice biophysical phenomena. In this case, under-filling of the FOV is permitted.

Spectral Radiance source selection

Where the measurement FOV is small, spectral radiance can be obtained from a measurement of spectral irradiance with an aperture placed at the plane of the luminous surface to define the FOV. In this case, the area on the surface of a sphere defined by the solid angle of the measurement can be approximated by the planar area over which spectral irradiance is measured.

Spectral Radiance Formula

The FOV solid angle can be computed from the FOV planar angle, γ, from:-

Spectral Radiance FOV solid angle Formula

Spectral radiance is then computed from:-

Spectral Radiance Diagram


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Category: Entrance Optics

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