ENE 451Fundamental 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.

Antireflection Coating

2 212 23 12 232 cosR A A A A

A12A23 > 0 and we want Rmin. cos = -1.

Antireflection Coating

2 1 3

2 2

22

22

4(2 1)

2 1

4

0: Very thin film

4

n n n

n tN

Nt

n

N

tn

n1 = 1.5, n3 = 1.7. What should be n2 for antireflection film?

Example

Find the thinnest film to be coated to prevent the reflected light give n1 = 1 and n3 = 3.6 if λ=0.83μm.

Example

Non-normal Incidence

qi

qi

qi

qt

qt

qt

A

B

C

D

E F

t

tn1

n2

Consider the case of non normal incidence as shown in the previous figure.

The emerging beam travels with an optical-path difference between them as

Non-normal Incidence

2 1

2 1 2

( )

( )

n AB BC n AD

n AE FC n AD n EB BF

Non-normal Incidence By Snell’s law,

and , this yields

Then we have

sin sin2t t

ACAE AG q q

sin iAD AC q

02f

nAE AD

n

0 2 f fn AD n AE n AE FC

Non-normal Incidence So that, an optical-path difference is

As EB = tcost , finally, we have

2f fn EB BF n EB

2 cosf tn t q

Non-normal Incidence Therefore, a round trip phase shift in this

case equals to

Therefore,

24 cos2 tn t q

2 2 212 23 12 23

4 cos2 cos tn t

R A A A A q

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.

Example

◦ (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).

Example

◦ (c) Calculate the reflectance for both s- and p-polarization for the case considered in (b).

Example

These are instruments which utilize coherent summation of wave amplitudes.

Two beam interferometer:

Interferometers

1 2

1 2

2 21 2 1 2 1 22 cos

i i

A

Ae A e

P A A A A

Mach-Zehnder Interferometer

2 2 21 2 1 2 1 2

2 2 21 2 1 2 1 2

( )

2 cos

2 cos

phase shift due to second baeam splitter

for light leaving from splitting in x- or z-direction.

x BSx

z BSz

BSx z

A A A A A

A A A A A

In general, BSx = + BSz Assume they are lossless beam splitters.

For 50:50 beam splitter.

Mach-Zehnder Interferometer

2 2 21 2 1 2 1 2

2 2 2 21 2

2 cos

2

x BSx

x z

A A A A A

A A A A

1 2

1 2

1 cos2

1 cos2

inAx BSx

inAz BSz

PP

PP

Mach-Zehnder Interferometer

1 1 2 21 2

1 1 2 21 2

2 2,

2

n L n L

n L n L

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.

Example

Michelson Interferometer

1 21 cos2in

out

PP

Michelson Interferometer

1 1 2 21 2

1 1 2 21 2

1 2 11

4 4,

4

2 for L2

n L n L

n L n L

n

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?

Example

Fabry-Perot Interferometer

After one round trip

After 2 round trips

Fabry-Perot Interferometer

2 2 2

1 1

optical loss coefficient

4 round trip phase shift

1 1

1 1

iin in

iin

i iin

A A R R Re e A R

nL

A A R Re e

A A R Re e R e e

After n round trips

Steady state (N )

Fabry-Perot Interferometer

0

1N

nin

n

i

A A R a

a Re e

0

22

2

1

1

1

1

1

1

Nn

n

in

in

aa

A RA

a

A RP A

a

Therefore,

Fabry-Perot Interferometer

2 22

2 2 2 2 2 2

2 2 2

2

2 2 2

1 1 cos sin

1 cos 2 cos sin

1 1 2 cos

1

1 2 cos

max for 2

min for 2 1

inout

out

out

a Re Re

R e Re R e

a R e R e

P RP

R e Re

P N

P N

If = 0 (lossless resonator), e- = 1

Fabry-Perot Interferometer

2

2max

2 2

2 2min

1

1 2

1 1

1 2 1

inout in

in inout

P RP P

R R

P R P RP

R R R

Fabry-Perot Interferometer

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?

Example

Soln

Example