Generalization of the geometric description of a light beam in radiometry and photometry

Lionel Simonot; Pierre Boulenguez

doi:10.1364/JOSAA.30.000589

Generalization of the geometric description of a light beam in radiometry and photometry

Lionel Simonot and Pierre Boulenguez

Journal of the Optical Society of America A
Vol. 30,
Issue 4,
pp. 589-595
(2013)
•https://doi.org/10.1364/JOSAA.30.000589

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Lionel Simonot and Pierre Boulenguez, "Generalization of the geometric description of a light beam in radiometry and photometry," J. Opt. Soc. Am. A 30, 589-595 (2013)

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Related Topics
Table of Contents Category
- Instrumentation, Measurement, and Metrology
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Etendue (080.2175)
- Photometry (120.5240)
- Radiometry (120.5630)

About this Article
History
- Original Manuscript: October 22, 2012
- Revised Manuscript: January 16, 2013
- Manuscript Accepted: January 21, 2013
- Published: March 6, 2013

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Abstract

Radiometric and photometric quantities rely on a geometric description of the beam subtended by a source and a receptor. In this paper, a generalization of this description is proposed as the product of the apparent size of the source times the receptor angular extent, whatever the natures of these elements: point, line, surface, or volume. The obtained flux density per geometric extent expressions are then applied to the determination of the irradiances induced in the near field and far field by a rectilinear source represented as a point source, a line source, and a surface source.

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Type ( $S / R$ )	Emitted Flux Density			Received Flux Density
Type ( $S / R$ )	Name		[unit]	Name		[unit]
Point		$Φ_{S}$			$Φ_{R}$
Point	Flux		[W]	Flux		[W]
Line		$d Φ_{S} / d L_{S}$			$d Φ_{R} / d L_{R}$
Line	N/A		[ $W \cdot m^{- 1}$ ]	N/A		[ $W \cdot m^{- 1}$ ]
Surface		$d^{2} Φ_{S} / d^{2} A_{S}$			$d^{2} Φ_{R} / d^{2} A_{R}$
Surface	Exitance		[ $W \cdot m^{- 2}$ ]	Irradiance		[ $W \cdot m^{- 2}$ ]
Volume		$d^{3} Φ_{S} / d^{3} V_{S}$			$d^{3} Φ_{S} / d^{3} V_{S}$
Volume	N/A		[ $W \cdot m^{- 3}$ ]	N/A		[ $W \cdot m^{- 3}$ ]

$j$	Far Field $Φ_{0 j} = \int_{X} Φ_{i j} d^{i} X$	Distributed Point Sources $Φ_{0 j, k} = Φ_{i j, k} Δ^{i} X_{k}$	Near Field	Name	[unit]
0	$Φ_{R} = Φ_{00}$	$Φ_{R} = \sum_{k = 1}^{N_{S}} Φ_{00, k}$	$Φ_{R} = \int_{X} Φ_{i 0, X} d^{i} X$	Radiant flux	[W]
1	$d Φ_{R} / d L_{R} = Φ_{01} \cos θ_{R} / ‖ S R ‖$	$d Φ_{R} / d L_{R} = \sum_{k = 1}^{N_{S}} Φ_{01, k} \cos θ_{R, k} / ‖ S_{k} R ‖$	$d Φ_{R} / {d L}_{R} = \int_{X} Φ_{i 1, X} \cos θ_{R, X} / ‖ S_{X} R ‖ d^{i} X$	N/A	$[W \cdot m^{- 1}]$
2	$d^{2} Φ_{R} / d^{2} A_{R} = Φ_{02} \cos θ_{R} / {‖ S R ‖}^{2}$	$d^{2} Φ_{R} / d^{2} A_{R} = \sum_{k = 1}^{N_{S}} Φ_{02, k} \cos θ_{R, k} / {‖ S_{k} R ‖}^{2}$	$d^{2} Φ_{R} / d^{2} A_{R} = \int_{X} Φ_{i 2, X} \cos θ_{R, X} / {‖ S_{X} R ‖}^{2} d^{i} X$	Irradiance	$[W \cdot m^{- 2}]$

Type ( $S / R$ )	Emitted Flux Density			Received Flux Density
Type ( $S / R$ )	Name		[unit]	Name		[unit]
Point		$Φ_{S}$			$Φ_{R}$
Point	Flux		[W]	Flux		[W]
Line		$d Φ_{S} / d L_{S}$			$d Φ_{R} / d L_{R}$
Line	N/A		[ $W \cdot m^{- 1}$ ]	N/A		[ $W \cdot m^{- 1}$ ]
Surface		$d^{2} Φ_{S} / d^{2} A_{S}$			$d^{2} Φ_{R} / d^{2} A_{R}$
Surface	Exitance		[ $W \cdot m^{- 2}$ ]	Irradiance		[ $W \cdot m^{- 2}$ ]
Volume		$d^{3} Φ_{S} / d^{3} V_{S}$			$d^{3} Φ_{S} / d^{3} V_{S}$
Volume	N/A		[ $W \cdot m^{- 3}$ ]	N/A		[ $W \cdot m^{- 3}$ ]

$j$	Far Field $Φ_{0 j} = \int_{X} Φ_{i j} d^{i} X$	Distributed Point Sources $Φ_{0 j, k} = Φ_{i j, k} Δ^{i} X_{k}$	Near Field	Name	[unit]
0	$Φ_{R} = Φ_{00}$	$Φ_{R} = \sum_{k = 1}^{N_{S}} Φ_{00, k}$	$Φ_{R} = \int_{X} Φ_{i 0, X} d^{i} X$	Radiant flux	[W]
1	$d Φ_{R} / d L_{R} = Φ_{01} \cos θ_{R} / ‖ S R ‖$	$d Φ_{R} / d L_{R} = \sum_{k = 1}^{N_{S}} Φ_{01, k} \cos θ_{R, k} / ‖ S_{k} R ‖$	$d Φ_{R} / {d L}_{R} = \int_{X} Φ_{i 1, X} \cos θ_{R, X} / ‖ S_{X} R ‖ d^{i} X$	N/A	$[W \cdot m^{- 1}]$
2	$d^{2} Φ_{R} / d^{2} A_{R} = Φ_{02} \cos θ_{R} / {‖ S R ‖}^{2}$	$d^{2} Φ_{R} / d^{2} A_{R} = \sum_{k = 1}^{N_{S}} Φ_{02, k} \cos θ_{R, k} / {‖ S_{k} R ‖}^{2}$	$d^{2} Φ_{R} / d^{2} A_{R} = \int_{X} Φ_{i 2, X} \cos θ_{R, X} / {‖ S_{X} R ‖}^{2} d^{i} X$	Irradiance	$[W \cdot m^{- 2}]$

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