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Fundamental of Optical Engineering
Lecture 9
The amount of light reflected when a beam
moves from one media to another can be
reduced by placing a thin coating layer
between them.
2
R A122 A23
2 A12 A23 cos
A12A23 > 0 and we want Rmin. cos = -1.
n2 n1n3
(2 N 1)
t2
2 N 1
4 n2 t2
4n2
N 0: Very thin film
t2
4n2
n1 = 1.5, n3 = 1.7. What should be n2 for
antireflection film?
Find the thinnest film to be coated to prevent
the reflected light give n1 = 1 and n3 = 3.6 if
λ=0.83μm.
D
qi
qi
qi
n1
t
A
C
qt
E
F
qt
n2
t
qt
B
Consider the case of non normal incidence as
shown in the previous figure.
The emerging beam travels with an opticalpath difference between them as
n2 AB BC n1 ( AD)
n2 AE FC n1 ( AD) n2 EB BF
By Snell’s law, AE AG sin qt
and AD AC sin qi , this yields
n0
2 AE AD
nf
AC
sin qt
2
Then we have
n0 AD 2n f AE n f AE FC
So that, an optical-path difference is
n f EB BF 2n f EB
As EB = tcosqt , finally, we have
2n f t cos qt
Therefore, a round trip phase shift in this
case equals to
2
4 n2t cos qt
Therefore,
4 n2t cos qt
R A A 2 A12 A23 cos
2
12
2
23
Consider a film of thickness t and refractive index 1.6
sandwiched between two media of refractive index 1.5.
◦ (a) determine all values of t for which the reflectance will be
a maximum at normal incidence for λ = 1 μm and calculate
the reflectance.
◦ (b) For an angle of incidence of 20 relative to the normal,
calculate the wavelength at which the reflectance will
maximum. Use the smallest value of t determined in (a).
◦ (c) Calculate the reflectance for both s- and p-polarization
for the case considered in (b).
These are instruments which utilize coherent
summation of wave amplitudes.
Two beam interferometer:
A1ei A2 ei
1
2
PA A12 A22 2 A1 A2 cos 1 2
Ax2 A12 A22 2 A1 A2 cos 1 2 BSx
Az2 A12 A22 2 A1 A2 cos 1 2 BSz
BSx ( z ) phase shift due to second baeam splitter
for light leaving from splitting in x- or z-direction.
In general, BSx = + BSz
Assume they are lossless beam splitters.
Ax2 A12 A22 2 A1 A2 cos 1 2 BSx
Ax2 Az2 2 A12 A22
For 50:50 beam splitter.
Pin
PAx
2
Pin
PAz
2
1 cos 1 2 BSx
1 cos 1 2 BSz
1
2 n1 L1
1 2
, 2
2 n2 L2
2 n1 L1 n2 L2
Suppose in a MZ interferometer for λ = 0.6328 μm, PAx = 0
and PAz = Pin. Then, a microscope slide 2 mm thick with a
reflective index of 1.55 is placed in one arm of the
interferometer. What are the new values of Pax and Paz.
Pin
1 cos 1 2
Pout
2
1
4 n1 L1
1 2
, 2
4 n2 L2
4 n1 L1 n2 L2
1 2 2 for L1
2n1
For a Michelson interferometer in air with λ = 1.06 μm, Pout =
0.5 Pin. One of the mirrors is displaced by increasing L1
continuously and Pout increases continuously to a final value
of 0.65 Pin. How large is the displacement?
After one round trip
A Ain 1 R R Re ei Ain 1 R
optical loss coefficient
round trip phase shift
A Ain 1 R 1 Re ei
4 nL
After 2 round trips
A Ain 1 R 1 Re ei R 2 e 2 e 2i
After n round trips
N
A Ain 1 R a n
n0
a Re ei
Steady state (N )
N
1
a
1 a
n 0
n
A
Ain 1 R
1 a
2
P A
Ain
2
1 R
1 a
2
1 a 1 Re cos
2
Re
2
sin
2
1 R 2 e 2 cos 2 2 Re cos R 2 e 2 sin 2
1 a 1 R 2 e 2 2 R cos e
2
Therefore,
Pout
Pin 1 R
2
1 R 2 e 2 2 Re cos 2
max Pout for 2 N
min Pout for 2 N 1
If = 0 (lossless resonator), e- = 1
Pout max
Pout min
Pin 1 R
2
1 R 2R
2
Pin 1 R
2
1 R 2R
2
Pin
Pin 1 R
1 R
2
2
Light from a laser of wavelength λ is transmitted through a
lossless Fabry-Perot interferometer in air. The mirror
reflectances are equal to R. As the mirror separation is
increased from an initial value, the transmitted power
increases to a maximum of 21 mW for a mirror separation D.
As the mirror separation is further increased D+0.25 μm, the
transmitted power decreases to a minimum of 0.3 mW.
(a) What is λ in μm?
(b) What is R?
(c) What is the transmitted power when the mirror separation
is D + 0.99 μm?
Soln