Biophotonics_Concepts_to_Applications

Example 4.2(a) At what angle from the normal in a lambertian LED is the

intensity level 50 % of the normal intensity? This is the half-power point.

(b) At what angle from the normal axis does the light appear only 40 % as

intense as it does when viewed down its centerline?

Solution:

(a) For a lambertian source the parameter g = 2 in Eq. (4.2). Using this

equation, at the half-power point IðÞ¼h I 0 cosh¼ 0 :50I 0. Thus

cosh¼0.50 so thatθ= 60°.

(b) In this case IðÞ¼h I 0 cosh¼ 0 :40I 0 , or cosh¼0.40 so thatθ= 67°.

4.1.4 Radiant Exposure or Radiant Fluence

Radiant exposureorradiantfluenceis the optical energy received by a surface per
unit area. Equivalently radiant exposure can be viewed as the irradiance of a surface
integrated over the time of irradiation. For example, for a light pulse the radiant
exposure is the average power of the pulse multiplied by the pulse duration.
A standard symbol for radiant exposure is H and the units are J/cm^2. The radiant
exposure also is known as theenergy density. For biophotonics applications H
ranges from approximately 1 mJ/cm^2 to 1 kJ/cm^2.

4.1.5 Radiance.

Radianceindicates how much of the optical power coming from an emitting or
reflecting surface will fall within a given solid angle in a specified direction. This
parameter is measured in units of watts per steradian per square meter (W/sr/m^2 ).
Thus knowing the radiance and the angle at which a tissue target is illuminated will
allow a determination of the total power incident on a tissue area. Radiance, des-
ignated by L, is defined by

L¼

d^2 P

dAdXcosh



P

AXcosh

ð 4 : 3 Þ

where
L is the observed or measured radiance (W/sr/m^2 ) in the directionθ
d is the mathematical differential operator
P is the total power (W) emitted
θ is the angle between the surface normal and the observation direction
A is the surface area (m^2 ) of the source
Ω is the solid angle (sr) subtended by the observation or measurement.

96 4 Fundamentals of Light Sources