Assignment 4

Problem 1:

Consider a point P on a line normal to the center of a disk. The point is a distance a from the center of the disk, which has a radius R.

(a) Show that the solid angle subtended by the disk at the point is

(b) Show that, when the point P is very far from the disk, the solid angle reduces to zero.

(d) Show that when R/a << 1 the expression for W simplifies to W = pR²/a².

(e) Explain physically what (b), (c), and (d) mean.

Problem 2:

(a) A lamp of 100 cd illuminates a book 4 m directly below it. Find the illuminance.

(b) A small-aperture spherical mirror is placed above the lamp at its focal distance of 3 m. The reflection coefficient of the mirror is 0.9. Find the increase in illuminance on the book.

Problem 3:

Assume that a quantum detector absorbs all the radiation falling onto it, and that each quantum of light releases 1 photoelectron. Use simple physical arguments to show that the responsivity of the detector is approximately 0.5 A/W in the visible region of the spectrum, where the photon energy is approximately 2 eV.

Problem 4:.

Consider a simple optical relay system consisting of two thin lenses of focal length f placed a distance 4f apart. The object plane is 2f in front of the first lens. The diameter of the first lens is D and the diameter of the second lens is 1.5 D.

Two important aspects of any imaging system are the amount of radiation passed by the system and the extent of an object that is seen by the system. One obtains knowledge of the first by calculating the aperture stop, entrance pupil and exit pupil and knowledge of the second comes from a calculation of the field stop, field of view, entrance window and exit window. The following is simply a recipe for calculating these quantities given a lens system.

Aperture stop, entrance pupil, and exit pupil.

Image all optical elements in the system into object space.

Find the angle subtended by each element image at the on-axis position of the object.

The element image with the smallest angle is the entrance pupil.

The physical object corresponding to this image is the aperture stop.

The image of the aperture stop in image space is the exit pupil.

(a) Find the location and the diameter of the aperture stop, entrance pupil, and exit pupil.

Field stop, entrance window, and exit window.

Image all optical elements in the system into object space.

Find the angle subtended by each image at the on-axis position of the entrance pupil.

The element image with the smallest angle is the entrance window

The physical object corresponding to this image is the field stop.

The image of the field stop in image space is the exit window.

(b) Find the location and the diameter of the field stop, entrance window, and exit window.

Field of View.

The angular field of view is the angle formed by the edges of the entrance window at the on-axis position of the entrance pupil.

In a system with vignetting, the angular field of view may also be defined as the largest angle of an input chief ray with the optical axis.

Problem 5:

Problem 6: