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Homework Set #1
Due: 1-15-14
This homework set does not cover lecture material (class just got started after all) but, instead,
reviews basic optics. If some of this material is new (not uncommon for graduate students these
days) and you need help, see me. The techniques required can be learned quickly and I need to
be able to depend on your knowing these concepts. With the exception of (2), these problems also
have a direct connection to ultrashort pulse laser systems.
1) An equilateral prism is made of glass with an index of refraction of 1.7. (a)
Find the input angle  such that the light travels parallel to the prism base
inside the prism. (b) What is the output angle of the light when it exits the
prism back into air?

A prism configured this way is said to be at “minimum deviation” and has
properties that are useful for some applications, including inside laser cavities.
Review: When light travels from one medium (#1) to another (#2) across a sharp interface, it refracts
according to Snell’s Law: n1 sin1 = n2 sin2. The angles are measured with respect to the normal to
the interface. The index of refraction of air is about 1.0003 and can usually be taken to be unity.
2) An arrow is followed by a diverging lens and a converging lens. The two lenses together form an
image.
(a)
Where is the final image? (Specify distance from the converging lens and whether it is to the
left or right of it.)
(b)
Is the final image real or virtual? Which way is the arrow’s image pointing: up or down?
(c)
What is the magnification of the system?
lens #1
f1 = -100 mm
50 mm
lens #2
f2 = 100 mm
150 mm
Review: If f is the focal length, o the distance to the object and i the image distance for a given lens:
1/f = 1/o + 1/I, with all distances measured from the center of the lens to good approximation for thin
lenses. Sign convention: the object distance is positive if the object lies before the lens and the image
distance is positive if it comes after. A positive image distance means the image is “real” – you can
see it on a card or screen. A negative image distance means the image is “virtual” – the output light
appears to be coming from the image, but a screen placed at the image will not show one. The
magnification m = -i/o. A negative magnification means the image is inverted. For multi-lens
problems you handle the first lens first, and then use the image of the first lens as the object for the
second lens. Note this means the second lens’s object can come after the second lens itself.
3) The beam expander. Two lenses are separated by a distance equal to the sum of their focal lengths.
A collimated beam (one consisting of, to good approximation, parallel rays) enters from the left with
beam diameter Din. The figure shows the situation for the cases where both lenses are positive and f2
> f1. (a) Show by ray tracing that the beam that exits the system is still collimated, but has a larger
diameter. (b) Show that Dout = (f2/f1) Din.
This is a commonly used configuration of lenses, sometimes called a telescope. If you work in a laser
lab, you’ll likely have to use one. These results hold even if one of the lenses is negative, so long as f1
+ f2 > 0.
Din
Dout
f1
f2
f1 + f2
4) Gratin
ng tutorial. A grating is an
a optical com
mponent that modulates ligght
spatiaally so that th
he outgoing diffracted
d
ligh
ht comes out at an angle thhat
depen
nds on its wavelength.
w
Gratings can
n be designeed so that tthe
diffraacted light is transmitted
t
th
hrough the grrating or, as sshown in Fig.. 1,
refleccted. Reflectio
on gratings are
a the most common.
c
Graatings are maade
by im
mposing a peeriodic, spatial modulatio
on of some pproperty of tthe
substrrate. The mo
odulated quan
ntity can be the absorptioon, reflectiviity,
transm
mission, thick
kness, or indeex of refractio
on. Althoughh somewhat leess
comm
mon, a substraate with spheerical curvatu
ure is sometim
mes used so tthe
gratin
ng can form
m images. Th
he first grating that stuudents usuallly
encou
unter in class is an array of slits on a sccreen. This is a transmissioon
gratin
ng and the mo
odulated quan
ntity is the trransmission (eeither 100% or
0%), as
a shown in Fig.
F 2.
Suppo
ose you havee monochrom
matic light in
ncident on a flat substratte,
reflecction grating. There will, in
n general, be many diffractted beams each
at a different angle,
a
called
d orders, coming
c
from
m the gratinng
nly one is sho
own in the fig
gures). The am
mount of pow
wer
simultaneously (on
y the choice of
o modulationn. A sinusoiddal
in eacch order is deetermined by
modu
ulation tends to
t put most of the power into the first oorder diffracted
beam (m=1), wherreas the mod
dulation show
wn in Fig. 3 ccan be used to
place more power is a high ordeer mode.
ng monochrom
matic light, tthe
Referrring back to Fig. 1, and still assumin
input and output raays are relateed via the gratting equationn, one variantt of
which
h is: d(sini + sind) = m . Here, d is the “wavvelength” of tthe
spatiaal modulation
n, called the groove
g
spacin
ng, i and d are the incideent
and diffracted anglles,  is the liight waveleng
gth and m is aan integer. Noote
that th
he angle of th
he incident an
nd diffracted rays
r
is measurred with respeect
to thee normal, as is usually th
he case in op
ptics. Differeent values off m
selectt the differentt possible outp
put angles or diffraction oorders. Note thhat
for th
he m = 0 ord
der, the gratting acts lik
ke a mirror independent of
wavellength.
d
i
d
Fig. 1
Incoming
wave.
A diffracted orderr.
Fig. 2. Grating formed ussing multiple slits..
(Picturre: http://blog.cenccophysics.com/
2009/007/diffraction-gratiing/)
d
Fig. 3. G
Gratings with a groove shape
like this are said to be blazzed.
(a) Explain
E
the sig
gn convention
n for the anglles in the grat
ating equationn. According to this conveention,
what
w is the sign
n of i and d for the case shown in Figg. 1? [Hint: coonsider the m=
=0 case.]
(b) A grating used
d in “Littrow” configuration operates sso that a speccified diffracttion order is retroreeflected back onto the inciident beam. Short
S
pulse lasser systems ffrequently em
mploy gratingss at or
neear Littrow, as do some narrow
n
line tunable
t
laserss, including ssome diode llasers. What is the
in
ncident angle for a Littrow
w grating with
h 500 groovess/mm operatinng in first ordder (m=1) witth 800
nm
m light?
(c) For short pulsee work, it is often
o
best to not permit orrders higher than m=1. Ennergy appeariing in
i usually wassted. Keeping
g the angle annd wavelengthh in (b), find a value for d tthat is
hiigher orders is
ju
ust sufficiently
y small to elim
minate orderss with m>1.