A certain light-emitting diode (LED) is centered at the origin with its surface in the xy plane. At far distances, the LED appears as a point, but the glowing surface geometry produces a far-field radiation pattern that follows a raised cosine law: that is, the optical power (flux) density in watts/m²is given in spherical coordinates by

where θ is the angle measured with respect to the direction that is normal to the LED surface (in this case, the z axis), and r is the radial distance from the origin at which the power is detected.

(a) In terms of P₀, find the total power in watts emitted in the upper half-space by the LED;

(b) Find the cone angle, θ₁, within which half the total power is radiated, that is, within the range 0 < θ < θ₁;

(c) An optical detector, having a 1-mm² cross-sectional area, is positioned at r = 1 m and at θ = 45°, such that it faces the LED. If one milliwatt is measured by the detector, what (to a very good estimate) is the value of P₀?

cos? e watts/m2 Pa = Po - ar 2nr2