Download Problem Set 3 Solutions - Holography and Diffractive Optics | ECE 527 and more Assignments Electrical and Electronics Engineering in PDF only on Docsity!
Homework 3 solutions
Opti527 – Fall 2008
Problem 1
An off-axis hologram is recorded with the geometry as shown below and then reconstructed
with a conjugate reference wave. Assume that the wavelength used during recording and
reconstruction is 488 nm, D
t
= 4 cm, Z
o
= 8 cm.
a) If the spatial frequency of the transparency varies from 0 to 1250 lines/mm and it is
illuminated by a normally incident plane wave determine the film diameter (D
f
) required to
capture all the light from the transparency. (Hint: consider the transparency to be a grating
with a period corresponding to 1250 lines/mm.)
b) If the film diameter is now fixed at D
f
= 4 cm determine the smallest angle θ that provides
non-overlapping spectral components of the images reconstructed with a conjugate
reference beam. Note any restrictions or assumptions.
c) Draw a figure of the reconstruction geometry and the spectra. What other consideration
should be taken into account?
Solution
a) From the grating equation we know that
max
sin sin ,
d i
m
f m
where f max
=1250 lines/mm and we are concerned with the first order m=1. Solving for the
diffracted angle we have that
1
max
sin 37.
d
θ λ f
−
and therefore the final width will be the original diameter plus 2*h, where
0
tan
d
f f
h Z
D D h cm
b) If on the other hand D f
=4cm then we can solve the reverse situation. From the grating
equation given in Eq. (1.1) we find,
and
and therefore that
max
B = f = =
a value that is undefined. This apparent error came from our req
overlapping spectral components, a requirement that is not consistent with the film diameter
constraint. We may instead assume that the object wave is weak, compared to the reference
wave, and we therefore have a less stringent r
c) The geometry look something like this:
where the blue lump in the spectrum plot is for the weaker object wave.
Problem 2
Write a ray tracing program (preferably in MATLAB or MATHCAD) using the algorithm
class for exact ray tracing. Generate a ray intercept plot for the following conditions: (Hologram
is 2 cm square. Evaluate the image at z
a) z
o
= -5 cm, z
r
= -20 cm, z
p
b) z
o
= -10 cm, z
r
= -∞, (x
p
= 175cm, y
c) z
o
= -5 cm, z
r
= -20 cm, z
p
Bergfield, Page
then we can solve the reverse situation. From the grating
we find, with m=1, that:
1
max
0
tan 26.
f
D
z
−
sin 26.
nm
Λ = = μ m
= = =916.42 lines/mm
which corresponds to a minimum angle
1 1
min
θ sin 3 B λ sin 1.34 ,
− −
a value that is undefined. This apparent error came from our requirement that we have non
overlapping spectral components, a requirement that is not consistent with the film diameter
constraint. We may instead assume that the object wave is weak, compared to the reference
wave, and we therefore have a less stringent requirement that
1
min
θ sin B λ 26.
−
c) The geometry look something like this:
where the blue lump in the spectrum plot is for the weaker object wave.
Write a ray tracing program (preferably in MATLAB or MATHCAD) using the algorithm
class for exact ray tracing. Generate a ray intercept plot for the following conditions: (Hologram
is 2 cm square. Evaluate the image at z
o
= -20.2 cm, λ
1
= λ
2
= 0.50 μm.
= 175cm, y
p
= 0 cm, z
p
= -10000cm), λ
1
= λ
2
= 0.50 μm.
= -20 cm, λ
1
= 0.50 μm, λ
2
= 0.501μm.
Bergfield, Page 2 of 4
then we can solve the reverse situation. From the grating
which corresponds to a minimum angle
uirement that we have non-
overlapping spectral components, a requirement that is not consistent with the film diameter
constraint. We may instead assume that the object wave is weak, compared to the reference
Write a ray tracing program (preferably in MATLAB or MATHCAD) using the algorithm given in
class for exact ray tracing. Generate a ray intercept plot for the following conditions: (Hologram
= 0.50 μm.
Solution
Many people decide to include the calculation of these factors into their code and that is fine.
From the notes we know that the paraxial image location is given by
0 0 0
3
0 0
0 0 0
3
0 0
0
3
0 0
p r p r r p
r p r p
p r p r r p
r p r p
p r
r p r p
x z z x z z x z z
x
z z z z z z
y z z y z z y z z
y
z z z z z z
z z z
z
z z z z z z
μ μ
μ μ
μ μ
μ μ
μ μ
And that the longitudinal magnification is given by
0
p r
M
z
z z
μ
μ
We can then proceed by plugging in the values given above:
a) Paraxial image location (x
3
,y
3
,z
3
) = (0,0,-5.012) cm for U3 and (0,0,9.950739) cm for U4, M=1.
b) Paraxial image location: (-0.175,0,-9.990) cm for U3 and (-.175,0,10.010) cm for U4, M=.
c) Paraxial image location: (0,0,-4.993) cm for U3 and (0,0,9.970) cm for U4, M=.
Problem 4
Determine the spherical and coma aberration coefficients for the conditions given in questions
2 a,b,c.
Solution
The spherical aberration coefficient (c.f. aber-3.pdf) is given by
2
3 3 2 2 2
0 0 0 0 0
r p r p r r r p
S
z z z z z z z z z z z z z
μ μ
μ μ
and the coma coefficient by
0 4
3 3 3 3
0 4
p r
x
p r
x x x x
C
z z z z
μ μ
Plugging the values for the parts a,b and c give us:
a) Spherical = -0.000056 (U3), .00673 (U4); Coma=0.00 (assuming on axis)
b) Spherical = 0.000003 (U3 and U4); Coma = 0.
c) Spherical = 0.000020 (U3), 0.006757 (U4); Coma = 0.
