Re: Solid Angles

[scherer@uiuc.edu (David Scherer)]

>G'day All,
>
>Could someone please explain the physical meaning of the unit:
>
>         Watts per square metre per steradian
>
>It's the "per steradian" which I am having trouble with.
>
>In case you are wondering, this question is related to LASER safety.
>
Physically, the source radiance is used because almost every optical source
is an extended source.  In the case of a laser with diameter D and
divergence half angle theta, the laser radiation is emitted into a solid
angle omega = pi theta^2.  Notice that the solid angle does not depend on
the beam diameter, only the divergence angle.

Aside for some math; skip if you wish.  To get this, back track the
divergent beam of diameter D to a virtual point source.  This distance is
D/(2 theta).  The solid angle is area (pi D^2 / 4) divided by (D/(2 theta))^2.

The total laser power is irradiated into this solid angle, but the
brightness (or radiance) depends on the beam size.  A larger source will be
spread out over more of the retina.  Of course, it is the energy density
(irradiance) at the retina that is important for safety purposes.  The
source radiance R (W/m^2-sr) can be related to the retinal irradiance I
(W/m^2), the pupil diameter d, occular focal length f, and occular
transmission T.

Another math aside:  The relationship is I= (pi) d^2 R T / (4 f^2).
See e.g. D.H. Sliney, et al, Amer. Ind. Hyg. Assoc. J., vol. 32, no. 7 (July
1971) pp. 415-431.

As one might guess, this is also useful for larger sources, such as diffuse
reflections or filament sources.

Dave Scherer
scherer@uiuc.edu