Lineare Optik, WiSe 2015/2016
Jun.-Prof. Dr. Thomas Weiss
Problem Set 7
due to 3/5 February
Problem 7.1: Some properties of lenses
a) Lenses can be converging or diverging. Is the focal length f of a converging lens positive or
negative? How do you construct the image of a diverging lense?
b) In the simplest case, both lens radii have the same absolute value but opposite sign. Therefore,
such lenses are either biconvex or biconcave. However, some lenses have one convex and one
concave interface. What happens if you turn such a lens around? Can you think of reasons to use
such lenses?
c) Consider a converging lens. We want to analyze the properties of the image depending on the
position of the object. Complete the following table:
Object
Distance
Image
Distance
Relative size
Upright?
Real?
∞ > g > 2f
g = 2f
2f > g > f
g=f
g<f
“Upright” means that the image has the same orientation as the object. “Real” images are images
that can be seen on a screen placed at the position of the image. So-called “virtual” images cannot
be imaged this way, but can be made visible by additional imaging systems such as additional
lenses or your eye. Therefore, they can be seen by looking through the lens.
d) You have an unknown converging lens and want to know its focal length. What quick test would
you use?
Problem 7.2: Optical systems beyond thin lenses
You have seen in the lecture how to describe a thin lens via the ABCD matrix formalism. One
important parameter to describe such a lens is the focal length. This can be expanded to arbitrary
imaging systems. In general, the focal length of an optical system described by an ABCD matrix is
given by f = −1/C.
a) A thick lens consists of two spherical surfaces that are separated by a distance d. Derive the
lensmaker’s equation for a thick lens made from a material with refractive index n in air.
b) Consider a system of two thin lenses with focal lengths f1 and f2 placed at a distance d. What is
the focal length of the combined system?
c) The interpretation of the resulting focal lengths is not as trivial as in case of the thin lens. For
example, a lens system with f1 , f2 > 0 has a negative focal length if d > f1 + f2 . Construct the
image of such a system (chose g such that the resulting image is real). Compare the properties of
the image to the table from Problem 1c) and discuss why it is impossible to describe such a lens
system by an effective converging lens.
Comment: You will hear more about the interpretation of the focal length of thick lenses and lens
systems when the problem sheet is discussed.
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Problem 7.3: Resonator stability
An optical resonator can be built from two spherical mirrors. It is described by the mirror radii R1
and R2 and their distance d. The resonator is called stable if the light that is reflected back and forth
in the cavity cannot escape.
a) We want to analyze the light in the cavity in a ray-optical picture. Show that you can replace the
cavity by an arrangement of two thin lenses, which is repeated infinitely. How are the focal lengths
fi related to the mirror radii Ri ? Write down the ABCD matrix M for one cavity round-trip.
Hint: Note that the ABCS matrix formalism does not distinguish between forward and backward
propagating rays. However, you have to take the propagation direction into account when you
analyze the sign of the radius of a curved surface.
b) The matrix M n describes the beam after n roundtrips. If the absolute values of the eigenvalues
λi of M exceed unity, M n diverges. The corresponding resonator is instable. It can be shown that
|λi | ≤ 1 is a sufficient criterion for stable resonators. Show that this is fulfilled if the eigenvalues
are complex and derive the stability criterion using only d and Ri .
Hint: The characteristic polynomial of a 2 × 2 matrix A is given by λ2 − Tr A λ + det A = 0.
Here, Tr denotes the trace of a matrix.
c) Is a Fabry-Perot cavity stable?
Problem 7.4: Abbe limit
The resolution of optical instruments depends on many parameters. Some limits are technical, such as
fabrication imperfections or aberrations. Other limits are physical. One of these limits is diffraction.
We will analyze its influence on the resolution of a microscope.
a) A microscope consists of two lenses. The first (so-called objective) lens creates a real image of
the object that is strongly magnified. The second lens (the so-called eyepiece) is placed such that
the resulting image is virtual. It can be seen by looking through the microscope, as discussed in
problem 1c). Draw the lens arrangement and the corresponding beam path of such a microscope.
b) Consider a point on the optical axis of the microscope and assume that this point emits light
in all directions. Some beams hit the objective and, therefore, contribute to the image seen in
the microscope. The outermost beams that hit the lens define the opening angle of the objective.
Derive an equation for the maximum opening angle.
Hint: The opening angle depends on the size of the lens.
c) Let’s now assume that we want to image a double slit (distance d between the slits) illuminated
from the back. The double slit diffracts the light. To extract information about d, at least the first
diffraction order must be considered. Therefore, the angle under which the first maximum occurs
must not exceed half of the maximum opening angle. The minimum slit distance that fulfills this
condition is defined as the resolution of the microscope. Derive an equation for this resolution in
dependence of the lens parameters and the incident light.
d) To increase the resolution of a microscope, one uses so-called immersion oil. This oil has a high
refractive index and is inserted between the objective and the object. Discuss, why this method
works.
General hint: The problems are designed to support the lecture. Some problems deal with topics that
have been only shortly introduced in the lecture. We recommend to use textbooks in addition to your
lecture notes to obtain a deeper understanding of these phenomena.
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