Projects - Cengage Learning

Appendix C C-63
denote the angles of incidence and refraction at the first surface, and angles
and w 2 denote the angles of incidence and refraction at the second surface.
Exercise 1
(a) Given r, s 1 , and s 2 , explain how to use a calculator to find u 1 and u 2 .
(b) Explain why w 1 u 1 , w 2 b u 2 , and w1 œ w2 œ b.
(c) Using Snell’s law and part (b) show that
sin u 1 n sin w1
œ
n sin w2 œ sin(b u 2 )
n sin w2 œ n sin(b w1)
œ
and conclude that
n sin(b w1) œ sin(b u 2 )
(4)
(d) Use an addition formula for sine and a Pythagorean identity to show that
n sin(b w1) œ (sin b)1n21 sin 2 w12 œ (cos b)(n sin w1)
œ (5)
Use equation (1) to obtain
n sin(b w1) œ (sin b) 2n 2 sin 2 u 1 cos b sin u 1 (6)
Then use equations (4) and (6) and an addition formula for sine to obtain
(sin b) 2n 2 sin 2 u 1 cos b sin u 1 sin b cos u 2 cos b sin u 2 (7)
(e) It is surprising that equation (7) can be be solved for b in a few steps.
Divide both sides by cos b and solve for tan b to derive the Fresnel lens
groove angle formula
w œ 2
(1)
(2)
(3)
tan b
sin u 1 sin u 2
2n 2 sin 2 u 1 cos u 2
(8)
Exercise 2 An optical designer must design a Fresnel lens 80 millimeters in
diameter to gather light from a lamp 100 mm from the lens and cause it to pass
through a small opening 400 mm from the lens on the side opposite the lamp.
The index of refraction of the lens material is 1.50. She lets s 1 100 mm and
s 2 400 mm, decides to make the width of each groove 0.4 mm, and calculates
that there will be 100 grooves. (How?)
She writes a computer program to generate a file containing the 100 groove
angles, in degrees, which will be the input to another computer program that will
generate commands for a computer-controlled machine (a lathe in this case) that
will produce the lens. Before trying to make the lens, she decides to calculate the
groove angle for grooves numbered 1, 25, 50, and 100, and use the calculated
values to check the output of her program. She makes a table with column headings
r in mm and u 1 , u 2 , and b in degrees. Her first r-value is 0.200 mm and her
second is 9.800 mm, and she calculates u 1 , u 2 , and b to four decimal places. Your
job as her assistant is to produce your own version of the table as another check
of her work. Use the results of Exercise 1 parts (a) and (e) to produce the table.