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Unformatted text preview: 21. Lecture 21
21.1 Diﬀraction and optical instruments
In the last lecture we saw that parallel rays after going through a slit are not parallel
anymore but have a range in angles from zero to θm = λ . Either experiment or a more
a
involved calculation show that, if the hole is round instead of a slit, then the minima
occurs at an angle
λ
θm 1.22 , circular hole
(21.1)
a
Since any optical instrument, for example a telescope has a hole through which light
comes in, even perfect parallel rays produce an image blurred as if they were not
actually parallel. For example in the simple case of a convergent lens shown in ﬁg.118,
horizontal parallel rays produce an image of size r given by
λ
r = 1.22 f
a (21.2) where f is the focal distance and a is the diameter of the lens. Namely, although rays
arrive parallel to the lens axis, after going through the lens diﬀraction spreads them
over a cone. In fact the image is a solid circle of radius r surrounded by diﬀraction rings.
These rings would make clear that the blurring is from diﬀraction and not from the
other aberrations we have already encountered. Since the wavelength λ is small this
blurring does not seem to be very important but we should remember that a telescope
further enlarges the image produced by the initial lens or mirror. Diﬀraction puts
a limit to how much this image can be enlarged because once we reach a resolution
where we can see the diﬀraction pattern, further enlargement would be useless. This is
sometimes called a theoretical limit, as opposed to other aberrations, nothing can be
done to improve it (other than making the telescope wider). Notice also that spherical
aberration requires using mostly the region of the lens close to its center. Restricting
the opening would accomplish that at the expense of increased diﬀraction.
One simple but very important device to consider is our own eye. Since the pupil
has a diameter of approximately 4mm then the image projected on the retina cannot
have more angular resolution than
θm 1.22 λ
400nm
1.22
10−4
4mm
4mm (21.3) in radians. This is quite small. It also implies that, if Nature was kind to us, we would
have enough cells in the retina to reach such resolution. A simple check is to consider,
at night, how far we would be able to see the two headlight of an incoming car as – 138 – separate sources. If the headlights are separated by 1m the distance at which we can
resolve them is
1m
D=
= 104 m = 10Km 6miles
(21.4)
θm
which is the right order of magnitude as we know from our driving experience. This
means that the human eye indeed reaches close to its theoretical resolution limit. To
improve we would need bigger eyes which would make focusing more complicated. Figure 118: Diﬀraction through the lens imply that perfectly parallel rays will give rise
to a blurred image even for a perfect lens. The eﬀect is very small but detectable when we
enlarge the image or look at it with enough resolution. This gives a theoretical limit to the
magniﬁcation of any optical device (in terms of its width). 21.2 Lightmatter interaction: Photoelectric eﬀect
Up to know we have studied the properties of light when it propagates through a
medium. However, light interacts with matter in many diﬀerent ways, for example it
is important to study how light is emitted an absorbed by matter. We start this study
by considering the photoelectric eﬀect. As we discussed at the beginning of the course,
electrons are free to move inside a metal, which explains why they are conductors.
When light shines on a metal these electrons can be ripped oﬀ the metal. This is
called the photoelectric eﬀect. Notice that electrons normally do not leave a metal
mainly because if they do so, the metal would be positively charged and would attract
them back. Light can kicked them out. In fact certain nightvision systems work in
this way by accelerating the electrons and amplifying the resulting current. From a
physical point of view we will be mainly concerned with the energy of the electrons
which are ripped oﬀ the metal. It turns out that the number of electrons coming out – 139 – is proportional to the intensity of light but their energy is proportional to frequency of
the light. This is rather surprising since the intensity of light describes precisely how
much energy reaches a certain area of the metal. It make sense that the total energy
transmitted to the electrons is proportional to the intensity. In fact it is because the
larger the intensity the larger the number of electrons. It is surprising however that
the energy of each individual electron depends only on the frequency of light and not
the intensity. The experimental result is described in ﬁg.119. No electrons emerge for
frequencies smaller than a cutoﬀ fc . For larger frequencies f the energy of the electrons
is given by
Eel. = h(f − fc )
(21.5)
The explanation of this fact was given by Einstein. He proposed that light is made out
of quanta which behave similarly as particles. Each quantum is called a photon and
has an energy given by
h
Eph. = hf =
ω = ω
(21.6)
2π
where h is a universal constant known as Planck’s constant. We also used the relation
between frequency and angular frequency ω = 2π f and deﬁned
= h
2π (21.7) This is an entirely new physical description. Planck had already observed that light is
emitted and absorbed in discrete amounts but Einstein took the photons as the real
picture of what light is made of. Now the explanation of the photoelectric eﬀect is very
simple. To extract an electron of the metal a minimal energy W of needed to overcome
the Coulomb attraction. This energy W is called the work function and depends on
the substance. If the frequency of the light is smaller that
fc = W
h (21.8) then a photon has not enough energy to kick out an electron. Two or more photons
would be needed but the probability of two photons hitting the same electron at the
same time turns out to be very small and can be ignored. If the photon has larger
energy than W the excess energy is transfered to the electron as kinetic energy:
Eph. = hf = W + Eel. = hfc + Eel.
from where eq.21.5 follows. – 140 – (21.9) Figure 119: Photoelectric eﬀect. Electrons are ejected from the metal by incident light. For
frequencies smaller that fc no electrons are ejected, for larger frequencies f , the energy of the
electrons is given by the formula Eel. = h(f − fc ) – 141 – ...
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This note was uploaded on 12/07/2011 for the course PHY 219 taught by Professor Na during the Fall '11 term at Purdue UniversityWest Lafayette.
 Fall '11
 NA

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