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    For the correct solving of illumination and light signals collection problems, it is useful to highlight some key points:

    • radiometry/photometry, are fields of optics that measure the electromagnetic radiation, which differ one each other only by the fact that photometry, in particular, deals with brightness as is perceived by the human eye;
    • the light source must be treated differently depending by if they are point or extended sources. In general it is always required to refer to the surface portion viewed by the observer. In particular in the case of point sources, the integral of the emitted radiation is distributed over 4π steradians (sr), while in the case extended surfaces is π sr.

    The energy of a single photon Wf (in Joules, where J = watts * sec) is given by:

    where h is the Planck's constant (6.62-10ˉ³ watt sec) and ν is the frequency in Hertz.

    where λ is expressed in μm.

    A plan surface, perpendicular to a radiation beam, is crossed by N photons in units of time given by:

    where P is the power beam and Wf is expressed in watt per second.

    The solid angle is measured in steradians (sr). One steradian is the solid angle subtended by a square of unitary area (eg. a squared meter), taken on the surface of a sphere of unitary radius (1 meter) (Fig. 4).

    Fig. 4 - Definition of steradian

    A sphere subtends 4π sr.


    The general expression of the solid angle Ω (in sr) is:

    where A is the area of the spherical surface and r is its radius.

    The inverse law of the squared distance from a point source states that the flux density is inversely proportional to the squared distance.
    The four most common geometric descriptions in radiometry are (radiometric terminology, photometric terminology):

    1. flux (radiant flux, luminous flux);
    2. irradiance (irradiance, illuminance);
    3. intensity (radiant intensity, light intensity);
    4. radiance (radiance, luminance).

    1. The flux (watts, lumen) is the total emitted power by a source, in all directions.

    2. The irradiance (watt · mˉ, lux) is the flux per unit of area of incident radiation on a surface. The irradiance describes the radiation impinging on a surface.

    3. Intensity (watts · srˉ, candle) is the flux per unit of solid angle emitted from a small source with respect to distance of the observer (point source).

    4. The radiance (w mˉ srˉ, lumen mˉ srˉ) is the intensity of an extended source per unit of area in the view direction, or, equivalently, can be defined as the emitted flow per unit of solid angle from a surface of unitary area viewed from the observation direction. The radiance describes the radiation radiated from a surface.

    Fig. 5 – Radiance calculation surfaces

    Light sources are characterized by:

    • The area of the emitting surface.
    • The power in watt (in radiometric units) or lumen (in photometric units). It is the total emitted flux (in all directions).
    • The flux density “Φ” (emittance in W·mˉ). It is the power divided by the area of emitting surface.
    • The intensity I (watt · srˉ) is the power divided by the emission solid angle. The angle, in steradians, must be the effective collected angle.
    • The radiance or luminance “L” (w mˉ srˉ, lumen mˉ srˉ), also called brightness, is the intensity divided by the observation area measured onto a perpendicular plane to the observation direction. If an emitting plane area is viewedfrom a not perpendicular direction, the area will be multiplied by the cosine of the angle between the observation direction andnormal direction to the surface.
    The radiance (or luminance) is constant (neglecting losses) along a beam. The radiance of the image can never exceed the radiance of the object itself.

    If an optical system forms the image of a source, if such image is smaller than the object, then its surface is smaller, but the angle under which it is formed is proportionately greater: the luminance remains constant (or less because losses).

    Fig. 6 - Radiance calculation surfaces

    From this follows that what it is important in a light source, when you need to make its image, it is its luminance and not its power or size. At equal luminance, two sources with different power have different surfaces.
    In order to increase the illumination (i.e. the irradiance W• mˉ - on the illuminated surface) of an object (the direction from whichthe light comes is not important) it can be used a greater power or greater surface source.

    In order to get the image of a light source by means an optical system, it is useless to increase the power or the surface of the source: it has to be increased the luminance of the source. When lamps equipped with rear collecting mirrors (usually parabolic or elliptical) are used, it is possible to gain energy by illuminating an object; this is impossible from images in which the angle is relevant (as in the case of source images formed by optical systems ).(*)

    Sometimes it is interesting to assess the amount of emitting energy from the surface of an object at a certain known temperature (i.e. the energy due to the only effect of the temperature). If the object is a "black body", for each temperature will be known the energy and the emitted spectrum. When the object cannot be considered a "black body" (as is usual), in order to know the energy and the spectrum of emitted radiation, it is necessary to know its emissivity. The emissivity is the ratio between the radiance of the object and the radiance of a blackbody at the same temperature and at the same wavelength.
    In general the emissivity of the surfaces and the emission direction are strongly influenced by the roughness of the surface: rough surfaces have higher emissivity (for the same material).

    (*) See also chap. 8 - "Coupling between optical systems".

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