R. J. Wilkes

Email: ph116@u.washington.edu

Physics 116

Session 24 Interference in gratings

and thin films Nov 8, 2011

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•! Exam 2 results

•! Avg 84

•! Std dev 15

•! Median 88

Grades should be on Webassign gradebook later today

3

Lecture Schedule (up to exam 3)

Today

10

Interference: reflected waves

•! Phase shift on reflection –! Recall behavior of wave on a rope:

•! If end of rope is rigidly secured, reflected wave gets inverted (180 deg phase change)

•! If end of rope can move up and down freely, no inversion

–! Similar behavior for reflection of EM waves if reflecting surface is on a medium with higher index of refraction (slower light propagation speed)

•! SO phase flip going from air to water, but no flip from water to air

•! Phase shift means interference can occur between incident and reflected waves

n2 = 1.5

n1 = 1.0

B

n’1 = 1.5

n’2 = 1.0

A

A’

C=180deg out of phase with A

C’

B’

C’ = in phase with A’

11

Interference in a thin layer of air

Tshim

t

Not to scale!

Glass is much

thicker than air layer

•! Air gap between flat glass plates –! Make a layer of air between 2 glass plates by inserting a shim (fine wire, or

hair) at both ends

•! Thickness of air layer is constant: t = Tshim

•! Ray reflected from top surface (air to glass): phase flip

•! Ray reflected from 2nd surface (glass to air): no flip (Ray 1)

•! Ray reflected from 3rd surface (air to glass): phase flip again (Ray 2)

•! Suppose angle of incidence is nearly vertical

–! so neglect the angle shown in drawing: Ray 2 travels 2t further than ray 1

•! Rays 1 and 2 show destructive or constructive interference if 2t is some multiple of ! wavelength

2 1

!2 ! !1 = 2t = m " / 2( ) Constr : m = 0,2, 4… Destr : m = 1, 3,5…

!2 ! !1 = 2t +"2

#$%

&'(= m

"2

#$%

&'(

)2t

"+1

2

#$%

&'(=m

2,

Constr if :m = 2, 4,6… Destr : m = 1, 3,5…

If there were no phase flip for ray 2, would be

Constructive if m= even (so full wavelength difference)

Destructive if m= odd (so half-wavelength difference) Phase flip of ray 2 in effect adds another half-wavelength:

12

Interference in reflected waves

•! Example: wedge of air between flat glass plates –! Make a wedge of air between 2 glass plates by inserting a shim (fine wire, or

hair) in one end

•! Thickness of air layer is proportional to x: t(x) = Tshim x

–! We see fringes as t increases: dark fringe when m is odd, bright when m is even

•! Requirements: –! Illuminate with monochromatic light, or fringes will be smeared in a rainbow!

–! Plates have to be very flat (“optical quality”) or ripples will make interference pattern impossible to see

•! Flat to within " wavelength: this is a very sensitive flatness test!

–! Light source and observer are “distant”

•! Meaning: ~ parallel light rays

x=0 x=1

Tshim

!2 ! !1 = 2t +"2

#$%

&'(= m

"2

#$%

&'(

)2t

"+1

2

#$%

&'(=m

2, Constr if :m = 2, 4,6… Destr : m = 1, 3,5…

www.oldham-optical.co.uk

13

Newton’s rings

•! Air gap between a convex lens and a flat glass plate –! Another air layer of variable thickness

–! This setup is used to test lenses – again, we can immediately see flaws at the scale of " wavelength

r

R

t

•! If you enjoy geometry and algebra, here’s how to determine radius r of the mth ring, for a given lens radius of curvature R and wavelength of light

•! Usually need a microscope to see rings for a typical lens

14

Thin films

•! Reverse the air gaps we’ve just discussed –! Make a thin film of some substance with n>nair

•! Soap bubble (water layer, held together by surface tension)

•! Oil slick on a puddle of water

•! Film of plastic material specifically designed for coating optical parts

–! Now ray 1 has a phase flip, and ray 2 does not

–! Otherwise, exactly the same analysis – except

•! Path difference occurs in medium with n>1

!n =!airn

, effective !1 =!air2

(just the phase flip), !2 = 2t (path length in film)

"! = !2 # !1, $ measure "! in units of wavelength to get phase relations

interference if !2

!n

%&'

()*#!1

!air

%&'

()*=

2t

!n

%&'

()*#

1

2

%&'

()*=m

2

$2t

!n#

1

2

%&'

()*=m

2, $

2nt

!air#

1

2

%&'

()*=m

2, Constr if :m = 0,2, 4,6… Destr : m = #1, +1, 3,5…

t

2 1

•! We have to take into account the wavelength in the medium (but we measure wavelengths in air

(nair ~ nvacuum =1.0)

•! Notice: if t is nearly 0, we still get destructive interference due to the phase flip for ray 1, so we need to add that case to our list – that’s why the -1 is in there

15

Antireflection coatings

•! Application of thin films: arrange for destructive interference of reflected rays –! Then film = non-reflective coating for lenses, glasses, etc

•! Example: glass (n=1.5 coated with magnesium fluoride (n=1.38)

What is the thickness of MgF layer needed? –! Now both ray 1 and ray 2 have a phase flip, but ray 2 also has path 2t

–! Choose wavelength in center of human-visible range: 555 nm

–! For destructive interference, we want m=1 (m=-1 means t=0)

!MgF =!air1.38

, effective !1 =!air2

(just the phase flip), !2 = 2t +!n2

(path length in MgF film plus phase flip)

"! = !2 # !1, $ measure "! in units of wavelength to get phase relations

interference if !2

!n

%&'

()*#!1

!air

%&'

()*=

2t

!n+

1

2

%&'

()*#

1

2

%&'

()*=m

2

2nt

!air

%&'

()*=m

2, Constr if :m = 0,2, 4,6… Destr : m = #1, +1, 3,5…

so t =1

2

!air2n

=1

4

!airn

%&'

()*=

555nm

4(1.38)= 100.5nm

MgF t

2 1

glass