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$: Www.bilkent.edu.tr/~ilday PHYS 415: OPTICS Review of Interference and Diffraction F. ÖMER ILDAY Department of Physics, Bilkent University, Ankara, Turkey$
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PHYS 415: OPTICS

Review of Interference and Diffraction

F. ÖMER ILDAY

Department of Physics,Bilkent University, Ankara, Turkey

I used the following resources in the preparation of almost all these lectures:Trebino’s Modern Optics lectures from Gatech (quite heavily used), andvarious textbooks by Pedrotti & Pedrotti, Hecht, Guenther, Verdeyen, Fowles and Das


Interference vs. Diffraction

• Interference is when we add up multiple but a finite number of E&M waves

• Diffraction is when we add up a continuum of E&M waves.

• Fundamentally, there is no difference.


Interference and Interferometers


Varying the delay on purpose

Simply moving a mirror can vary the delay of a beam by many wavelengths.

Since light travels 300 µm per ps, 300 µm of mirror displacement yields a delay of 2 ps. Such delays can come about naturally, too.

Moving a mirror backward by a distance L yields a delay of:

2 L /cDo not forget the factor of 2!Light must travel the extra distance to the mirror—and back!

Translation stage

Input beam E(t)

E(t–)

Mirror

Output beam


We can also vary the delay using a mirror pair or corner cube.

Mirror pairs involve tworeflections and displace the return beam in space:But out-of-plane tilt yieldsa nonparallel return beam.

Corner cubes involve three reflections and also displace the return beam in space. Even better, they always yield a parallel return beam:

“Hollow corner cubes” avoid propagation through glass.

Translation stage

InputbeamE(t)

E(t–)

MirrorsOutput beam

[EdmundScientific]


The Michelson Interferometer

Beam-splitter

Inputbeam

Delay

Mirror

Mirror

Fringes (in delay):

*1 2 0 1 0 2

2

2 1 1 2 0 0

Re exp ( 2 ) exp ( 2 )

2 Re exp 2 ( ) ( / 2)

2 1 cos( )

outI I I c E i t kz kL E i t kz kL

I I I ik L L I I I c E

I k L

since

L = 2(L2 – L1)

The Michelson Interferometer splits a beam into two and then recombines them at the same beam splitter.

Suppose the input beam is a plane wave:

Iout

L1

where: L = 2(L2 – L1)

L2 Outputbeam

“Bright fringe”“Dark fringe”


The Michelson Interferometer

Beam-splitter

Inputbeam

Delay

Mirror

Mirror

The most obvious application of the Michelson Interferometer is to measure the wavelength of monochromatic light.

L = 2(L2 – L1)

Iout

L1

L2 Outputbeam

2 1 cos( ) 2 1 cos(2 / )outI I k L I L


Huge Michelson Interferometers may someday detect gravity waves.

Beam-splitter

Mirror

Mirror

L1

L2

Gravity waves (emitted by all massive objects) ever so slightly warp space-time. Relativity predicts them, but they’ve never been detected.

Supernovae and colliding black holes emit gravity waves that may be detectable.

Gravity waves are “quadrupole” waves, which stretch space in one direction and shrink it in another. They should cause one arm of a Michelson interferometer to stretch and the other to shrink.

Unfortunately, the relative distance (L1-L2 ~ 10-16 cm) is less than the width of a nucleus! So such measurements are very very difficult!

L1 and L2 = 4 km!


The LIGO project

A small fraction of one arm of the CalTech LIGO interferometer…

The building containing an arm

The control center

CalTech LIGO

Hanford LIGO


The LIGO folks think big…

The longer the interferometer arms, the better the sensitivity.

So put one in space, of course.


Suppose the input beam is not monochromatic(but is perfectly spatially coherent):

Iout = 2I + c Re{E(t+2L1 /c) E*(t+2L2 /c)}

Now, Iout will vary rapidly in time, and most detectors will simply integrate over a relatively long time, T :

/ 2 / 2

1 2

/ 2 / 2

( ) 2 Re ( 2 / ) *( 2 / )

T T

out

T T

U I t dt U IT c E t L c E t L c dt

The Michelson Interferometer is a Fourier Transform Spectrometer

The Field Autocorrelation!

Beam-splitter

DelayMirror

L1

L2

2 Re ( ') *( ' 'U IT c E t E t dt

Recall that the Fourier Transform of the Field Autocorrelation is the spectrum!!

Changing variables: t' = t + 2L1 /c and letting = 2(L2 - L1)/c and T


Fourier Transform Spectrometer Interferogram

The Michelson interferometer output—the interferogram—Fourier transforms to the spectrum.

The spectral phase plays no role! (The temporal phase does, however.)

Inte

grat

ed ir

radi

ance

0 Delay

Michelson interferometer integrated irradiance

2/

1/

Frequency

Inte

nsity

Spectrum

A Fourier Transform Spectrometer's detected light energy vs. delay is called an interferogram.


Fourier Transform Spectrometer Data

Interferogram

This interferogram is very narrow, so the spectrum is very broad.

Fourier Transform Spectrometers are most commonly used in the infrared where the fringes in delay are most easily generated. As a result, they are often called FTIR's.

Actual interferogram from a Fourier Transform Spectrometer


Michelson-Morley experiment

19th-century physicists thought that light was a vibration of a medium, like sound. So they postulated the existence of a medium whose vibrations were light: aether.

Michelson and Morley realized that the earth could not always be stationary with respect to the aether. And light would have a different path length and phase shift depending on whether it propagated parallel and anti-parallel or perpendicular to the aether.

Mirror

Supposed velocity of earth through the aether

Parallel and anti-parallel propagation

Perpendicular propagation

Beam-splitter

Mirror


Michelson-Morley Experiment: Results

Michelson and Morley's results from A. A. Michelson, Studies in Optics

Interference fringes showed no change as the interferometer was rotated.

The Michelson interferometer was (and may still be) the most sensitive measure of distance (or time) ever invented and should’ve revealed a fringe shift as it was rotated with respect to the aether velocity.

Their apparatus


The “Unbalanced” Michelson Interferometer

Now, suppose an object isplaced in one arm. In additionto the usual spatial factor, one beam will have a spatiallyvarying phase, exp[2if(x,y)].

Now the cross term becomes:

Re{ exp[2if(x,y)] exp[-2ikx sinq] }

Distorted fringes (in position)

Place anobject inthis path

Misalign mirrors, so beams cross at an angle.

z

x

Beam-splitter

Inputbeam

Mirror

Mirror

q

exp[if(x,y)]

Iout(x)

x


The "Unbalanced" Michelson Interferometercan sensitively measure phase vs. position.

Phase variations of a small fraction of a wavelength can be measured.

Placing an object in one arm of a misaligned Michelson

interferometer will distort the spatial fringes.

Beam-splitter

Inputbeam

Mirror

Mirror

q

Spatial fringes distorted by a soldering iron tip in

one path


The Mach-Zehnder Interferometer

The Mach-Zehnder interferometer is usually operated “misaligned” and with something of interest in one arm.

Beam-splitter

Inputbeam

Mirror

Mirror

Beam-splitter

Output beam

Object


Mach-Zehnder Interferogram

Nothing in either path Plasma in one path


The Sagnac Interferometer

The two beams take the same path around the interferometer and the output light can either exit or return to the source.

Beam-splitter

Inputbeam

Mirror

Mirror

Mirror

Beam-splitter

Inputbeam

Mirror

Mirror

It turns out that no light exits! It all returns to the source!


Why all the light returns to the source in a Sagnac interferometer

For the exit beam:

Clockwise path has phase shifts of p, p, p, and 0.

Counterclockwise path has phase shifts of 0, p, p, and 0.

Perfect cancellation!!

For the return beam: Clockwise path has phase shifts of p, p,p, and 0. Counterclockwise path has phase shifts of 0, p, p, and p. Constructive interference!

Beam-splitterInput

beam

Mirror

Mirror

Reflection off a front-surface mirror yields a phase shift of p (180 degrees).

Reflective surface

Reflection off a back-surface mirror yields no phase shift.

Exit beam

Returnbeam


The Sagnac Interferometer senses rotation

Suppose that the beam splitter moves by a distance, d, in the time, T, it takes light to circumnavigate the Sagnac interferometer.

As a result, one beam will travel more, and the other less distance.

If R = the interferometer radius, and W = its angular velocity:

Thus, the Sagnac Interferometer's sensitivity to rotation depends on its area. And it need not be round!

2

0 0

20

exp( ) exp( )

sin ( )

outI E ikd E ikd

I kd

2

20

(2 / ) 2 ( ) / 2 /

sin (2 / )out

d R T R R c R c c

I I k c

Area

Area

qR

qWT

d

d = Rq

Sagnac Interfer- ometer


Newton's Rings

Get constructive interference when an integral number of half wavelengths occur between the two surfaces (that is, when an integral number of full wavelengths occur between the path of the transmitted beam and the twice reflected beam).

This effect also causes the colors in bubbles and oil films on puddles.

You see the colorl when: 2L = ml

L

You only see bold colors when m = 1 or 2. Otherwise the variation withl is too fast for the eye to resolve.


Newton's Rings


Multiple-beam interference: The Fabry-Perot Interferometer or Etalon

A Fabry-Perot interferometer is a pair of parallel surfaces that reflect beams back and forth. An etalon is a type of Fabry-Perot interferometer, and is a piece of glass with parallel sides.

The transmitted wave is an infinite series of multiply reflected beams.

Transmitted wave:

Incident wave: E0

Reflected wave: E0r

d = round-trip phase delay inside medium = 2kL

2 2 2 2 2 2 2 2 30 0 0 0 0( ) ( ) ...i i i

tE t E t r e E t r e E t r e E

Transmitted wave: E0t

r, t = reflection, transmission coefficients from glass to air2

0t E2 2

0it r e E

2 2 20( )it r e E

2 2 30( )it r e En nair = 1nair = 1

L


The Etalon (cont'd)

The transmitted wave field is:

2 2 2 2 2 2 2 2 30 0 0 0 0

2 2 2 20

( ) ( ) ...

1 ( ) ( ) ...

i i it

i i

E t E t r e E t r e E t r e E

t E r e r e

2 20 0 / 1 i

tE t E r e

2

1

1 sin / 2T

F

2 2

2

2 2

1 1

r rF

r R

The transmittance is:

2 22 40

2 2 20 1 (1 )(1 )t

i i i

E t tT

E r e r e r e

4 2 2 2 2

4 4 2 2 2 4 2 2

(1 ) (1 )

{1 2cos( )} {1 2 [1 2sin ( / 2)]} {1 2 4 sin ( / 2)]}

t r r

r r r r r r

where:

Dividing numerator and denominator by 2 2(1 )r


Etalon Transmittance vs. Thickness, Wavelength, or Angle

The transmittance varies significantly with thickness or wavelength.We can also vary the incidence angle, which also affects d (via L).

As the reflectance of each surface (r2) approaches 1 (F increases), the widths of the high-transmission regions become very narrow.

Transmission maxima occur when d/ 2 = mp:

2pL/l = mp

or:

/ 2L m

2

1

1 sin / 2T

F

2 4 /kL L

Tra

nsm

ittan

ce


Does this look familiar?

Recall that a finite train of identical pulses can be written:

where g(t) is a Gaussian envelope over the pulse train.( ) {III( / ) ( )} ( )E t t T g t f t

( ) {III( / 2 ) ( )} ( )E T G F

g(t) = exp(-t/t)The light field trans-mitted by the etalon!

The peaks become Lorentzians.


The Etalon Free Spectral Range

lFSR =Free Spectral

Range

The Free Spectral Range is the wavelength range between transmission maxima.

4 4 4 42 2

[1 / ]FSR FSR

L L L L

24 [1 / ] 1 /4 1 12

2 2FSR FSR

FSR

LL

L L

lFSR

2 4 /kL L

Tra

nsm

ittan

ce


Etalon Linewidth

The Linewidth dLW is a transmittance peak's full-width-half-max (FWHM).

Setting d equal to dLW/2 should yield T = 1/2:

2

1

1 sin / 2T

F

2 21 sin / 2 / 2 2 or sin / 4 1/LW LWF F

2/ 4 1/ 4 /LW LWF F

4 /L

2 1

2LW

R

L r

22

1

rF

R 2

4 12LW

L R

r

2

4LW LW

L

Substituting and we have:

Or:

For d << 1, we can make the small argument approx: l

lLW

Tra

nsm

ittan

ce

The linewidth is the etalon’s wavelength-measurement accuracy.


The Interferometer or Etalon Finesse

The Finesse is the number of wavelengths the interferometer can resolve.

2

22

1 12

rLR R

L r

F

/ [1 ]R FTaking 1r

The Finesse, F , is the ratio of the Free Spectral Range and the Linewidth:


How to use an interferometer to measure wavelength

1. Measure the wavelength to within one Free Spectral Range using a grating or prism spectrometer to avoid the interferometer’s inherent ambiguities.

2. Scan the spacing of the two mirrors and record the spacing when a transmission maximum occurs.

3. If greater accuracy is required, use another (longer) interferometer with a FSR ~ the above accuracy (line-width) and with an even smaller line-width (i.e., better accuracy).

Interferometers are the most accurate measures of wavelength available.


Anti-reflection Coating

Notice that the center of the round glass plate looks like it’s missing.It’s not! There’s an “anti-reflection coating” there (on both the front and back of the glass).


Anti-reflection Coating Math

Consider a beam incident on a piece of glass (n = ns) with a layer of material (n = nl) if thickness, h, on its surface.

It can be shown that the Reflectance is (for such thin media, we need to go back to Maxwell’s equations):

2 2 2 2 2 20 0

2 2 2 2 2 20 0

( ) cos ( ) ( ) sin ( )

( ) cos ( ) ( ) sin ( )l s s l

l s s l

n n n kh n n n khR

n n n kh n n n kh

Notice that R = 0 if:2

0l sn n n

/ 2 / 4kh h At normal incidence, and if (i.e., )2 2

02 2

0

( )

( )s l

s l

n n nR

n n n


Multilayer coatings

Typical laser mirrors and camera

lenses use many layers.

The reflectance and transmittance

can be tailored to taste!


Stellar interferometry

Taken from von der Luhe, of Kiepenheuer-Institut fur Sonnenphysik, Freiburg, Germany.

Stars are too small to resolve using normal telescopes, but interferometry can see them.

Stellar interferometers operate in the radio (when the signals are combined electronically) and visible (where the beams are combined optically).


“Photonic crystals” use interference to guide light—sometimes around corners!

Interference controls the path of light. Constructive interference occurs along the desired path.

Augustin, et al., Opt. Expr., 11, 3284, 2003.

Yellow indicates peak field regions.

Borel, et al., Opt. Expr. 12, 1996 (2004)


Convolution


The Convolution

The convolution allows one function to smear or broaden another.

( ) ( ) ( ) ( )

( – ) ( )

f t g t f x g t x dx

f t x g x dx

changing variables:

x t - x


The convolution can be performed visually.

Here, rect(x) * rect(x) =

(x)


Convolution with a delta function

Convolution with a delta function simply centers the function on the delta-function.

This convolution does not smear out f(t). Since a device’s performance can usually be described as a convolution of the quantity it’s trying to measure and some instrument response, a perfect device has a delta-function instrument response.

( ) ) ( ) ( – )

( )

f t t a f t u u a du

f t a


The Convolution Theorem

The Convolution Theorem turns a convolution into the inverse FT of the product of the Fourier Transforms:

Proof:

{ ( ) ( )} = ( ) ( )f t g t F G wF

{ ( ) ( )} ( ) ( – ) exp( )

( ) ( ) exp(– )

( ){ ( exp(– )}

( ) exp(– ) ( ( (

f t g t f x g t x dx i t dt

f x g t x i t dt dx

f x G i x dx

f x i x dx G F G

F


The Convolution Theorem in action

2

{ ( )}

sinc ( / 2)

x

k

F{rect( )}

sinc( / 2)

x

k

F

rect( ) rect( ) ( )x x x

2sinc( / 2) sinc( / 2) sinc ( / 2)k k k


The symbol III is pronounced shah after the Cyrillic character III, which is

said to have been modeled on the Hebrew letter (shin) which, in turn,

may derive from the Egyptian a hieroglyph depicting papyrus plants

along the Nile.

The Shah Function

The Shah function, III(t), is an infinitely long train of equally spaced delta-functions.

t

III( ) ( )m

t t m


The Fourier Transform of the Shah Function

If = 2n, where n is an integer, the sum diverges; otherwise, cancellation occurs. So:

{III( )} III(t F

) exp( )

)exp( )

exp( )

m

m

m

t m i t dt

t m i t dt

i m

t

III(t)

F {III(t)}

2


Fraunhofer Diffraction


Diffraction

Light does not always travel in a straight line.

It tends to bend around objects. This tendency is called diffraction.

Any wave will do this, including matter waves and acoustic waves.

Shadow of a hand

illuminated by a

Helium-Neon laser

Shadow of a zinc oxide

crystal illuminated

by aelectrons


Why it’s hard to see diffraction

Diffraction tends to cause ripples at edges. But poor source temporal or spatial coherence masks them.

Example: a large spatially incoherent source (like the sun) casts blurry shadows, masking the diffraction ripples.

Untilted rays yield a perfect shadow of the hole, but off-axis rays blur the shadow.

Screenwith hole

A point source is required.


Diffraction of a wave by a slit

Whether waves in water or electromagnetic radiation in air, passage through a slit yields a diffraction pattern that will appear more dramatic as the size of the slit approaches the wavelength of the wave.

slit size

slit size

slit size


Diffraction of ocean water waves

Ocean waves passing through slits in Tel Aviv, Israel

Diffraction occurs for all waves, whatever the phenomenon.


Diffraction Geometry

We wish to find the light electric field after a screen with a hole in it.

This is a very general problem with far-reaching applications.

What is E(x1,y1) at a distance z from the plane of the aperture?

Incident wave This region is assumed to be

much smaller than this one.

x1x0

P10

A(x0,y0) y1y0


Diffraction Solution

The field in the observation plane, E(x1,y1), at a distance z from the aperture plane is given by a convolution:

0 0

1 1 1 0 1 0 0 0 0 0

( , )

011 0 1 0

01

2 2201 0 1 0 1

( , ) ( , ) ( , )

exp( )1 ( , )

A x y

E x y h x x y y E x y dx dy

ikrh x x y y

i r

r z x x y y

where :

and :

A very complicated result! And we cannot approximate r01 in the exp by z because it gets multiplied by k, which is big, so relatively small changes in r01 can make a big difference!


Fraunhofer Diffraction: The Far Field

0 0

2 21 1

1 1 0 1 0 1 0 0 0 0

( , )

exp( ), exp exp ,

2A x y

x yikz ikE x y ik x x y y E x y dx dy

i z z z

Let D be the size of the aperture: D 2 = x02 + y0

2.

When kD2/2z << 1, the quadratic terms << 1, so we can neglect them:

0 0

2 22 20 1 0 1 0 01 1

1 1 0 0 0 0

( , )

( 2 2 ) ( )exp( ), exp exp ,

2 2 2A x y

x x y y x yx yikzE x y ik ik E x y dx dy

i z z z z

Recall the Fresnel diffraction result:

This condition corresponds to going far away: z >> kD2/2 = D2/If D = 100 microns and = 1 micron, then z >> 30 meters, which is far!


Fraunhofer Diffraction Conventions

As in Fresnel diffraction, we’ll neglect the phase factors, and we’ll explicitly write the aperture function in the integral:

1 1 0 1 0 1 0 0 0 0 0 0, exp ( , ) ( , )ik

E x y x x y y A x y E x y dx dyz

This is just a Fourier Transform!

Interestingly, it’s a Fourier Transform from position, x0, to another position variable, x1 (in another plane). Usually, the Fourier “conjugate variables” have reciprocal units (e.g., t & , or x & k). The conjugate variables here are really x0 and kx = kx1/z, which have reciprocal units.

So the far-field light field is the Fourier Transform of the apertured field!

E(x0,y0) = constant if a plane wave


The Fraunhofer Diffraction formula

where we’ve dropped the subscripts, 0 and 1,

, exp ( , ) ( , )x y x yE k k i k x k y A x y E x y dx dy

E(x,y) = const if a plane wave

Aperture function

We can write this result in terms of the off-axis k-vector components:

kx = kx1/z and ky = ky1/z

, ( , ) ( , )x yE k k A x y E x y F

kz

ky

kx

x = kx /k = x1/z and y = ky /k = y1/z

and:

or:


Fraunhofer Diffraction from a slit

Fraunhofer Diffraction from a slit is simply the Fourier Transform of a rect function, which is a sinc function. The irradiance is then sinc2 .


Fraunhofer Diffraction from a Square Aperture

The diffracted field is a sinc function in both x1 and y1 because the Fourier transform of a rect function is sinc.

Diffractedirradiance

Diffractedfield


Diffraction from a Circular ApertureA circular aperture

yields a diffracted

"Airy Pattern,"

which involves a

Bessel function.

Diffracted field

Diffracted Irradiance


Diffraction from small and large circular apertures

Recall the Fourier scaling!

Far-field intensity pattern

from a small aperture

Far-field intensity

pattern from a large aperture


Fraunhofer diffraction from two slits

A(x0) = rect[(x0+a)/w] + rect[(x0-a)/w]

0 x0a-a

1 0( ) { ( )}E x A x F

1 1

1 1

sinc[ ( / ) / 2]exp[ ( / )]

sinc[ ( / ) / 2]exp[ ( / )]

w kx z ia kx z

w kx z ia kx z

1 1 1( ) sinc( / 2 ) cos( / )E x wkx z akx z kx1/z

w w


Diffraction from one- and two-slit screens

Fraunhofer diffraction patterns

One slit

Two slits


Diffraction from multiple slits

Slit DiffractionPattern Pattern

Infinitely many equally spaced slits yields a far-field pattern that’s the Fourier transform


Young’s Two Slit Experiment and Quantum Mechanics

Imagine using a beam so weak that only one photon passes through the screen at a time. In this case, the photon would seem to pass through only one slit at a time, yielding a one-slit pattern.

Which pattern occurs?

Possible Fraunhofer diffraction patterns

Each photon passes

through only one slit

Each photon passes

through both slits


Fresnel Diffraction


Fresnel Diffraction: Approximations

2 22 20 1 0 10 1 0 11 1

12 2 2 2

x x y yx x y yz z

z z z z

2 2

2 22 0 1 0 101 0 1 0 1 1

x x y yr z x x y y z

z z

But, in the denominator, we can approximate r01 by z.

And, in the numerator, we can write:

0 0

2 2

0 1 0 11 1 0 0 0 0

( , )

1( , ) exp ( , )

2 2A x y

x x y yE x y ik z E x y dx dy

i z z z

This yields:

1, 1 1 / 2 But if


Fresnel Diffraction: Approximations

Multiplying out the squares:

0 0

2 2 2 20 0 1 1 0 0 1 1

1 1 0 0 0 0

( , )

( 2 ) ( 2 )1, exp ( , )

2 2A x y

x x x x y y y yE x y ik z E x y dx dy

i z z z

0 0

2 22 20 1 0 1 0 01 1

1 1 0 0 0 0

( , )

( 2 2 ) ( )exp( ), exp exp ,

2 2 2A x y

x x y y x yx yikzE x y ik ik E x y dx dy

i z z z z

This is the Fresnel integral. It's complicated!

It yields the light wave field at the distance z from the screen.

Factoring out the quantities independent of x0 and y0:


Diffraction Conventions

We’ll typically assume that a plane wave is incident on the aperture.

0 0( , )E x y const

And we’ll explicitly write the aperture function in the integral:

And we’ll usually ignore the various factors in front:

2 22 2

0 1 0 1 0 01 11 1 0 0 0 0

( 2 2 ) ( )exp( ), exp exp ( , )

2 2 2

x x y y x yx yikzE x y ik ik A x y dx dy

i z z z z

2 2

0 1 0 1 0 01 1 0 0 0 0

( 2 2 ) ( ), exp ( , )

2 2

x x y y x yE x y ik A x y dx dy

z z

It still has an exp[i(t – k z)], but it’s constant with respect to x0 and y0.


Fresnel Diffraction: Example

Fresnel Diffraction from a single slit:

Far from

the slit

zClose to the slit

Incident plane wave

Slit


Fresnel Diffraction from a Slit

This irradiance vs. position emerges from a slit illuminated by a laser.

x1

Irra

dia

nce


Diffraction Approximated

The approximate intensity vs. position from an edge:

Such effects can be modeled by measuring the distance on a “Cornu Spiral”

But most useful diffraction effects do not occur in the Fresnel diffraction regime because it’s too complex.

For a cool Java applet that computes Fresnel diffraction patterns, try http://falstad.com/diffraction/

www.bilkent.edu.tr/~ilday phys 415: optics review of interference and diffraction f. Ömer ilday...

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