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EE466 Optical Engineering
Homework 2 & 3
1. In the simple thin lens-retina model of the unaided eye, let’s suppose the distance
between the lens and retina is 18mm. What diopter power is required for an
emmotropic (normal) eye to focus clearly on a distance object?
2. If the unaided eye has a stronger lens with a power of +=62 D, the light focuses
before it reaches the retina, and the 18-mm eye is myopic. What is the far point for
this eye?
3. Determine the near points and the far points for the following eyes, given that each
can provide a maximum accommodation of 8D.
a. An emmetropic eye (normal eye sight)
b. A myopia eye which has a lens power that is 5D greater than an emmetropic
eye.
c. A hyperopic eye with a lens power that is 5D less than an emmetropic eye.
4. A Galilean Telescope is constructed using two lenses one with a positive focal length
and one with a negative focal length. If the focal length of one of the lenses is
f=30cm what is the focal length of the second lens such that the angular magnification
is 10x? Draw the ray trace for this telescope.
5. A rough object is illuminated with Nd:YAG laser at a wavelength of 1.06 m and 1W
optical power. Assume that the object scatters light with equal intensity in the entire
 



hemisphere  1    I r 2 sin  d d  . At a distance of 100 meters away from the
0 0


rough object a lens is used to collect the light. The lens has diameter of 10mm and a
focal length of f=50mm.
a. What is the total optical power collected by the lens (assume that 100% of the
light is scattered by the object and there is no absorption or reflection off the
light by the lens)?
b. Using the paraxial approximate what is the diameter of the spot on the focal
plane of the lens. (You should use the concept of field of view to determine
this.)
A

0
6. Show that the wave equation  2  k 2 U r   0 , becomes  T2 A  j 2k
z
 2
2
2 
  T 
 , if the field is divided into a spherical wave front, e  jkz , and a

2
2 
x
y 

A



 k .
slowly varying envelope, Ar  , resulting in U r   Ar  e  jkz , where
z


7. A helium-neon laser emits a Gaussian beam (=633nm) with a total power of 10mW
and e-2 power radius of 1mm. (a) What is the intensity of the laser at the surface of the
moon at a distance of 376,100km? (b) What is the laser intensity on the moon if the
initial beam is expanded to an e-2 radius of 1m?
8. (Problem 3.1-6 from the book) The light emitted from a Nd:YAG laser at a
wavelength of 1.06 m is a Gaussian beam of 1W optical power and beam divergence
2o=1 milliradian. Determine the beam waist radius, the depth of focus, the
maximum intensity, and the intensity of the beam axis at a distance z=100 cm from
the beam waist.
9. A simple digital camera has a single lens with a focal length of f=50mm, an aperture
diameter of 35mm, and a CCD detector with 1024X1024 pixels and a total size of 1/3
inch. (Assume that there is no spacing between the individual pixels.) A HeNe laser
(=633nm) with an e-2 full width of 1 mm is located 1km away from the camera.
a. How many pixels does the laser illuminate if the camera is looking directly at
the beam?
b. If the laser is changed to 1m away, how many pixels does the laser illuminate?
Figure 1. Free space optical communications link.
Figure 1 shows a free space optical communication link that consists of a laser transmitter
and detector. The lens on the transmitter is designed to change the Gaussian beam waist.
The lens on the receiver is designed to collect the light and focus it onto a detector. The
laser has a power of Pt=5mW, a wavelength of =670nm, the receiver needs to collect
Prec=10W to attain the required signal to noise ratio, and the separation between the
transmitter and receiver is L=100m. In a free-space optical communication link there is a
trade-off between the lens sizes and required alignment accuracy. There are two required
alignments. (1) The pointing of the transmitter towards the receiver. (2) The pointing of
the receiver towards the transmitter.
10. If the beam waist at the transmitter is Wo=1mm, what is the required collection
diameter D?
11. With the transmitter beam waist of Wo=1mm, what is the transmitter alignment error
for which the intensity at the collection lens drops by 50%?
12. Because of errors in the imaging quality of a lens, the ratio between the lens diameter
and the focal length needs to remain below a certain value. If the maximum diameter
of the collection lens is 4D<f and the detector width is 2mm, what is the maximum
pointing error of the receiver for the lens diameter calculated in problem 12?