Photonics and Lasers

Appendix A - Solid Angle and the Brightness Theorem

Appendix A

 

Solid Angle and the Brightness Theorem

SOLID ANGLE

 

The solid angle is the 2-D analog of the conventional 1-D angle , as illustrated in Fig. A-1. Just as the angle is defined as the distance along a circle divided by the radius of that circle, so the solid angle is analogously defined as the area on the surface of a sphere divided by the radius squared of that sphere. The units for and are radians (r) and steradians (sr), respectively, although it should be noted that both of these measures of angle have no actual dimensions. Since the total surface area of a sphere is A R2, the total solid angle in one sphere is 4 sr.

For situations with symmetry about an axis (such as an optical fiber or the normal to a plane surface), the two types of angles can be easily related. Figure A-2 shows a differential area dA on the surface of a sphere, in the form of a thin ring centered about the symmetry axis. This ring can be thought of as the intersection of the spherical surface with two cones, one of half-angle , and other of half-angle + d . The width of this ring is R d , and the radius of the ring is R sin . The differential solid angle is then

AppA_Photonics_and_Lasers-1.gif

The solid angle inside a cone of half-angle can be determined by integrating

AppA_Photonics_and_Lasers-2.gif


AppA_Photonics_and_Lasers-3.gif

Figure A-1 The solid angle is defined analogously to the conventional angle , using the fraction of a sphere's area rather than the fraction of a circle's circumference.

 

AppA_Photonics_and_Lasers-4.gif

Figure A-2 Geometry for relating differential solid angle d to the differential change in cone angle da around an axis of symmetry.

 

It is often of interest to consider the small-angle approximation, where « 1. In this limit, cos 1 - 2/2, so

AppA_Photonics_and_Lasers-5.gif

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