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Unformatted text preview: Physics 4510 — Optics Fall 2007 Homework Set #3  due Friday, October 12, 5:00 pm. 1. The telescope depicted below consists of two spherical mirrors. The radius of
curvature is 2.0m for the larger mirror (which has a hole through its center) and
60cm for the smaller. How far from the smaller mirror should the ﬁlm plane be
located if the object is a star? What is the effective focal length of the system? 2. In a human eye, the image is formed on the retina, about 23 mm behind the
lens. Compute the approximate size of the image of the moon as cast on the
retina. The moon has a diameter of 2160 miles and is abdut 230,000 away. For the matrix problems in this set, write your ray vector as [y] (and not Rim]
(1 y as I did in class) and use the handout on matrices on the course website. Note
that with ray vectors defined this more standard way, it is no onger always true
that det (M)= 1. But det (M)=1 for a system that begins and ends with the ray
traveling through the same value of n. (For instance, travel through air, n=1, go
into glass, come back out, travel through air, n=1). 3. Check to make sure the matrix labeled “distance of length d” in the handout is
correct. 4. A concaveplanar glass (n= 1.50) lens in air has a radius of 10.0cm and a thickness
of 1.00cm. Determine the system matrix and check that its determinant is 1. At what
positive angle (in radians measured above the axis), should a ray strike the lens at a
height of 2.00m, if it is to emerge from the lens at the same height but parallel to the
Optical axis? Do not use thin lens approximation. Instead, combine matrices for the two interfaces and the thick glass. 5. Let’s do some integrals! A plane wave with wavevector ko/z‘falls on a
slit of width D. We can approximate the electric field at a value of 2 just beyond the slit as
i ﬁx»)
.1. i
g i“
D E(x) = ED. for x< D/2
' g = O, for xl > D/2 a) If the probability for finding a photon at position x is P (x) o: E2, what is the r.m.s.
spread in x, 6 x, for the photons? b) Calculate E(kx), the Fourier transform of E(x) c) If the probability of ﬁnding a photon with pX = h kx is P (px=h kx)oc  E(kx) 2 what is the
r.m.s. spread in px? d) Hopefully you will find your answer to problem 0) somewhat unphysical. Speculate
as to how my setup of this problem may have led you to this sorry state. I I . )\ _
in any case, we see that a square envelope is a very diffractional object. a
6. A 10 watt argon—ion laser emits light at 4800A , with a minimum spotsize (w) of
2mm, located at z=0. a) How far will the beam travel before the spotsize is 4mm?
b) What fraction of the power (not the electric ﬁeld) of the beam passes through a disc
of radius w(z), centered on the axis? 7. Starting with the equation in the handout for a highermode Gaussian beam, Emp,
ed at _
derive an expression for the phase velocity Vp= aﬂi/é—‘g along the aXIS, as function of z, m and p. Show that the phase velocity is greater than c. A Gaussian beam with wavelength is incident on a lens of focal length f located at 2:8.
Calculate the focal length of the lens such that the output is focused on the sample a
z=t+L. (Remember that when a Gaussian beam comes to a focus, R=oo. Do n_ot
assume it, L>>Zo,. Le, at: + g; =l= j; ! Hint: If you find yourself trying to decide what root of a quadratic equation to take,
consider that in the limit of t, L» 20. you should recover g; +2: a i. You will need to do more math manipulation for this problem than is usual for my
homeworks. Courage! ...
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 Fall '06
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