Solutions Chapter 4
4.1
In order for a 6 ft tall man to see himself from head to toe, how tall a mirror needs to be
placed at the following distances from the man?
a.
b.
c.
4.2
6 ft away
10 ft away
20 ft away
What are the glass numbers for
a.
b.
c.
NBK7
SF11
LaSFN9
a) NBK7: 517642
b) SF11: 785258
c) LaSFN9: 850322
4.3
Which glass has a higher dispersion in the visible spectrum, crown or flint?
Flint glasses have lower Abbe numbers and therefore higher dispersion than crown
glasses.
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4.4
For a penta prism:
a.
b.
Make a layout of the unfolded diagram to get the PPP
If the input surface of the penta prism is 1 cm wide, what is the total OPL, if the
glass index is 1.5?
a)
1cm
1cm
2 cm
1cm
b) The diagram above shows that the total path through the glass is 1cm+1cm+ 2 cm =
3.41cm. The 2 was found by the Pythagorean theorem.
Total optical path length = tn = (3.41cm)1.5 = 5.12cm
4.5
The new “Back-Saving Horizontal Reading Glasses” for watching TV while lying down
are being sold. As a student of geometrical optics, studying prisms, you should be able to
answer the following questions.
a.
What is the optical diagram for using a right angle prism to view the TV?
b.
How do these eyeglasses work, especially considering parity and handedness of the
image?
a)
b) How does this configuration work considering handedness and parity?
The image is right side up, but the single reflection causes the parity to be reverted
left and right. Words are written backwards from the way they would appear on the
screen. A more complex system would allow the glasses to work as show below by
providing two reflections so that the parity is correct:
Half reflecting side
Mirror
Side
Mirror
Side
Deviation is not 90 degrees
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Half the light is lost every time the hypotenuse is reached
2
4.6
What is the parity for an image which has traversed a:
a.
b.
c.
penta prism
dove prism
corner cube
a)
b)
c)
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4.7
Calculate the angle of deviation for F, d and C light for a prism made of SF4 (755276)
glass with a 22 apex angle and 30 angle of incidence.
4.8
Derive the expression for angle of deviation for a thin prism.
4.9
What is the handedness (parity), and orientation of an image seen through the following
prisms? How will the image be changed when looking though the prism at the letter R?
a.
b.
c.
Abbe Prism
Penta Prism
Right Angle Prism
(1) Light reflected by the hypotenuse
(2) Light reflected at the two smaller faces (Porro)
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4.10
What is the correct orientation and description of the R in the final space?
4.11
A student has two glass rods. Rod A is # 300800 glass and rod B is # 900400 glass.
a.
b.
c.
What is the ratio of velocities of light in glass (A) to glass (B)?
In which rod is the velocity of light slower, (A) or (B)?
Which is the crown glass?
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4.12
There are two pieces of glass, each 10 cm in length. Glass A has a glass # of 200700 and
Glass B is 600350.
a.
What is the ratio of the velocity of light in glass A to glass B?
b.
Which glass has the higher speed of light?
c.
Which is the flint glass?
4.13
Plot the deviation angle () for a prism made of F2 with apex angle of 25 versus
incidence angle (I) using d-wavelength light.
a.
What is the minimum deviation for d-light?
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b.
4.14
What is the minimum deviation for d-light if we assume a thin prism?
For SF11 glass, what is the index of refraction calculated to five decimal places for the
following wavelengths:
a.
b.
c.
d.
400 nm
500 nm
600 nm
700 nm
Using Matlab to calculate n:
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4.15
For the following optical systems, draw the orientation of the letter at the observation
plane.
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4.16
What is the minimum refraction index that a right angle prism must have to reflect the
light by 90 with Total Internal Reflection (TIR)?
4.17
For the drawing below using an isosceles glass prism (n = 1.5) the apex angle (2.5) is
such that the ray is parallel to the axis inside the prism. What is the optical path length
for the ray? What is it along the axis?
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4.18
What are the Abbe numbers and glass numbers for:
From the Schott Catalog:
a.
b.
c.
d.
e.
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SF59
SF59
F14
F14
BK1
BK1
BaK4
BaK4
FK54
FK54
20.4
38.2
63.5
56.0
90.7
glass number
952204
601382
510635
569560
437907
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4.19
A crown (BaK4) thin prism with an apex angle of 15 is to be combined with a flint
prism (SF12) so as to produce no net deviation for d-light
a.
b.
Find the apex angle for the contact flint prism.
Find the angular deviation for C-light for this prism combination.
a)
b)
4.20
A hollow (and empty) 60 apex angle is immersed in a liquid of index 1.74. What is the
angle of minimum deviation?
4.21
A glass prism with n = 4.62 shows an angle of minimum deviation of 48.2. What is the
apex angle?
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4.22
A 5 ft tall lady would like to buy a mirror to use while dressing. She requires that it be
large enough so that she can see her shoes and hair simultaneously. If she sets the mirror
at distances of: (a) 3 ft, (b) 5 ft, and (c) 10 ft away, what is the smallest mirror she should
obtain for each case?
This problem is similar to problem 4.1, in which a 6ft tall man wants to see himself from toes to
eyes. In this case, we have a 5ft tall woman. A ray leaving her shoes must be reflected back to
her eyes and must also meet the mirror at its bottom. Also, a ray that leaves her hair must be
reflected straight back to her head at the top of the mirror. Therefore, due to geometry, the
mirror must be half of the woman’s height for this to occur, or 2.5ft tall, regardless of distance.
4.23
A Pechan prism has the following side view:
a)Determine the handedness of the image and indicate whether or not the image is inverted
The image is inverted because with the odd number of bounces ( 5 bounces) the handedness
is left handed. Input:
Output:
b) Draw the Tunnel diagram (unfold the prism)
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The tunnel diagram is shown below. This is drawn by unfolding the prism system at each
reflecting surface
c) Can you see any applications for this prism?
Applications for a prism that inverts an image, but has zero angular deviation, may be for
rifle scopes to flip the image. There is a niche market of these prisms to guards who practice
a rare form of yoga where they do headstands for extended periods of time, but must
maintain a clear, upright view from the guard station in order to fulfill their watch duties.
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4.24
Plot the deviation angle () versus incidence angle (I1) for a prism with an apex angle of
50, made of NBK7 glass, using d-light. At what angle does the minimum deviation
occur?
(degrees)
0
deviation
-10 0
20
40
60
80
100
-20
-30
-40
-50
incident angle
Minimum deviation occurs about 20 degrees incident
4.25
A prism with an apex angle of 90 is shown in the figure below. A ray enters this prism
at 30 and exits at 50. What is the index of refraction of the prism?
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4.26
Given a prism with an apex angle A = 6.3 and an Ohara Glass number of S-FPL51, what
is the angle of minimum deviation, min, for C, d, and F-light?
4.27
Plot the index of refraction vs. wavelength (F, d, C only) for NBK7, N-FK5, and F5
glasses. Plot all points on the same graph and use wavelength in nm. Use the equation
K1 
K2
2
for NBK7 and solve for K1 and K2.
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4.28
Describe the dispersion for the following glasses as either higher or lower than KF3
glass. (KF3 glass has an Abbe # of about 55). Classify each type as a crown or a flint:
a.
b.
c.
d.
4.29
N-FK5
NBK7
F5
N-LaSF45
For a prism made of F4 glass with an apex angle of 25, plot the deviation angle ()
versus incidence angle (I) using d-wavelength light, C-light and F-light.
a.
What is the angular spread of C to F light rays for an incident angle of 35?
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b.
c.
d.
What is the angular spread of d to F light rays for an incident angle of 35?
What is the minimum deviation for d light?
What is the minimum deviation for d light if we assume a thin lens?
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4.30
For a 3" prism made of NBK7, with a 3” base and height of 4”, compare the dispersion at
F, C and d light. (Hint: Use K2 from problem 4.27).
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4.31
For the following deviating prisms, draw the correct orientation of the object, R.
4.32
The index of refraction for a glass at three different wavelengths is shown is below:
a.
b.
λ (nm)
n
486.1
587.6
656.3
1.525
1.517
1.514
What is the Abbe #?
What is the glass number?
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4.33
Given a right-angle prism with an apex angle of 7, what is the angle of deviation for a
prism made from:
a.
b.
c.
NBK7
F4
NSF57
δ = -A( n-1 )
a) N-BK7
b) F4
c) N-SF57
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A= 7o = 0.122 rad
n = 1.5168
n = 1.6166
n = 1.847
δ = 0.630 rad , 3.612 deg
δ = 0.753 rad , 4.316 deg
δ = 0.103 rad , 5.929 deg
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4.34
4.35
This prism stack is used to focus optical radiation onto a solar cell, which is 2 mm
diameter (assume rotational symmetry). Where should the solar cell be located from the
reference surface to achieve maximum signal if :
a.
All of the prisms are made of n = 1.5 glass, and:
(1) Prism 1 has an apex angle of 5
(2) Prism 2 has an apex angle of 3
(3) Prism 3 has an apex angle of 1
b.
All of the prisms have an apex angle of 5, but each prism has a different index:
(1) Prism 1 (nd =1.477)
(2) Prism 2 (nd = 1.286)
(3) Prism 3 (nd = 1.0955)
For the following prism types, what is the parity (odd or even)? Make an unfolded
(Tunnel) diagram:
a.
b.
c.
Penta Prism
Porro Pair Prisms
Abbe Prism
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4.36
The index of refraction for sapphire (Al2O3) for the ordinary and extraordinary rays is
( in micrometers):
no 2  1 
ne 2  1 
a.
b.
1.4313493 2 0.65054713 2 5.3414021 2


 2  0.00528  2  0.014238  2  325.0178
1.5039759 2
0.55069141 2 6.5927379 2


 2  0.005480263  2  0.014802  2  402.8951
Plot the index versus wavelength from 400 nm to 5 μm in wavelength.
Over this range, what wavelength has the greatest index difference?
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4.37
A pulse of white light which is 1ps wide incidence of a glass fiber optic (nF = 1.6,
nd = 1.55, nC = 1.5). The output pulse needs to have the red light (C) separated from the
blue light (F) by 1 ns. How long must the fiber be made?
The problem sets up a situation where light is normally incident on a piece of glass and
the dispersion of the glass will cause the light to “smear” out 1000 times its incident
width. Since blue light sees a higher n, it will trail red light by 1 nanosecond. Setting up
the expression to find this distance, all we need is the OPD between red and blue and
divide by the speed of light to get the optical time difference.
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