A diffuse reflectance calibration panel used to check a spectrometer behaves as a…

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Geometrical optics and imaging · Geometrical optics and imaging

Optics and Photonics, Ch Radiometry and photometry

Problem

A diffuse reflectance calibration panel used to check a spectrometer behaves as a Lambertian surface of radiance

L = 50

W m

^{-2}

^{-1}

, the same in every direction above it. A relay lens, air on both sides (

n=1

), collects a narrow beam from the panel whose marginal ray has height

h_1 = 4

mm and half-angle

u_1 = 0.03

rad before the lens, and forms an image with transverse magnification

M = 6

. ((a)) Starting from the hemisphere integral

M_e = \int_0^{2\pi}\int_0^{\pi/2} L\cos\theta\sin\theta\,d\theta\,d\phi,

derive

M_e = \pi L

and hence find the radiant exitance of the panel. ((b)) Using the Lagrange invariant

\mathcal H = nhu

from Chapter 1 (preserved because every elementary ray-transfer matrix has determinant 1) and the etendue of a narrow circular beam,

G = n^2A\Omega = \pi^2\mathcal H^2

, derive that a lossless lens leaves the beam's etendue unchanged. Find the marginal ray's half-angle

u_2

after the lens and confirm numerically that the etendue is the same on both sides.