PY3101 Optics - Physics | University College CorkDepartment of Physics PY3101 Optics Diffraction...

$: PY3101 Optics - Physics | University College CorkDepartment of Physics PY3101 Optics Diffraction condition Note that the condition for constructive interference is asinθ=mλ. ColáistenahOllscoileCorcaigh,$
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M.P. Vaughan

PY3101 Optics

Diffraction

Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland

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Department of PhysicsPY3101 Optics

Overview

• Near and far-field diffraction• Diffraction through a single slit

• Diffraction limited imaging

• Multiple slit diffraction

• The diffraction condition

• Diffraction gratings• Reflection gratings

• Monochromators

• Diffraction around objects

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Diffraction regimes: near and far-field


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What is diffraction?

• Diffraction is the ‘bending’ of waves around objects or through

apertures

• It is an interference effect

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Diffraction – water waves


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Augustin-Jean Fresnel

Augustin-Jean Fresnel (1788 –1827)

Augustin-Jean Fresnel

Extensive work on interference

and diffraction

The wave theory of light

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Double slit diffraction

Thomas Young (1773 – 1829)

Double-slit experiment

First conclusive evidence of the

wave nature of light

Thomas Young

Double slit model may be derived from the N-slit case putting N = 2, so not explicitly derived here.


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Fresnel-Poisson-Arago bright spot

Predicted by Poisson as a counter-example to try to disprove Fresnel.

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Light passing through a narrow aperture

Huygens-Fresnel Principle


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Light passing through a narrow aperture

Maximum possible path difference

.max DABBPAP ==−=∆

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Limiting cases: λλλλ >> D

∆max always less than λ – wavelets add constructively in all directions.

Emergent field looks like point source.


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Limiting cases: λλλλ << D

Both constructive and destructive interference outside shaded region

Wavelets add constructively in this region

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Fresnel and Fraunhofer diffraction

Near field (Fresnel diffraction)

Far field (Fraunhofer diffraction)


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• Fresnel diffraction• Diffraction pattern varies with increasing distance from aperture

• Fraunhofer diffraction• Diffraction pattern settles down to a constant profile

• Applies for when the radial distance from the aperture satisfies the Fraunhofer condition

(see later).

Fresnel and Fraunhofer diffraction

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Fraunhofer diffraction: single slit


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Single slit diffraction

ELis the field

strength per unit length

EPis the total

field a the point P

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Field at x

.dxEdEL

=

Contribution to field EPdue to dE

( )( )[ ] .sin dxxkrt

xr

EdE

L

P−= ω

Total field EP

( )( )[ ] .sin

2/

2/∫− −=D

D

L

Pdxxkrt

xr

EE ω


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r(x) is given by the cosine rule

( ) ( )θπ −−+=2

222 cos2RxxRxr

x

or

( ) .sin2

1

2/1

2

2

−+= θR

x

R

xRxr

To find a closed form solution, we must approximate this expression.

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Taylor series expansion of r(x)

The Taylor series expansion for a function (1 + ξ)1/2 is

( ) K+−+=+82

112

2/1 ξξξ

Hence,

( )

++−= Kθθ 2

2

2

cos2

sin1R

x

R

xRxr

and

( ) .cos2

sin 22

K++−= θθR

kxkxkRxkr


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The Fraunhofer condition

The third term in the expression for kr(x) takes its maximum when x ± D/2 and θ = 0. That is

.48

cos2 2

2

2

22

2

R

D

R

kD

R

kx

λπ

θ =→

The condition that this term makes a negligible contribution to the phase is

.4 2

2

πλπ

<<R

D

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The Fraunhofer condition

Neglecting the factor of 4 in the denominator of the condition just found, it may be re-written as

.DR

D λ<<

This is the Fraunhofer condition for far field diffraction.


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Far field approximations

Assuming that the Fraunhofer condition is valid, the third term in the expression for kr(x) may be neglected and we have

( ) .sinθkxkRxkr −≈

The 1/r(x) factor appearing in the integral for EPis less

sensitive to changes in r(x) than the phase and we may simply put

( ).11

Rxr≈

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Integrating over x

Using these approximations, the expression for the total field EPbecomes

To perform this integral, we note that

[ ] ( ){ }.Imsinsin sinθωθω kxkRtiekxkRt

+−=+−

[ ] .sinsin2/

2/∫− +−=D

D

L

PdxkxkRt

R

EE θω


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The total field EP

Integrating over the x-dependent part

where

.sin2

θβkD

=

,sin

sin

2/

2/

sin2/

2/

sin

ββ

θ

θθ

Dik

edxe

D

D

ikxD

D

ikx =

=

−−∫

Hence, the total field EPis

( ).sinsin

kRtR

DEE

L

P−= ω

ββ

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Intensity profile for a single slit

Averaging EP over time gives

The squared modulus of this will be proportional to the intensity, i.e.

.sin

2 ββ

R

DEE

L

P=

( ) ( ) .sin

0

2

ββ

θ II =


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The zeros of the peaks occur at values of

where m is an integer. Hence, the first zeros around the central peak are given by

,sin2

πθβ mkD

==

.sinD

λθ =

Note that this result is only valid for λ < D. In other cases, there are no zeros from –π to π.


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Diffraction from a circular aperture

The analysis of Fraunhofer diffraction from a circular slit follows similar lines to that of a single slit. The result for the intensity is

( ) ( ) ( ),

sin20

2

1

=

c

cJ

IIβ

βθ

where

( ),2sin θβ kDc=

D is the diameter of the aperture and J1 is the first order Bessel function.

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Bessel functions

( ) ( ) .2

1 sin∫−−−=

π

π

ττ τπ

dexJxni

n


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Intensity profile for a circular aperture

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The Airy disc

Airy pattern by Sakurambo. A computer-generated image of an Airy disk.URL: http://en.wikipedia.org/wiki/File:Airy-pattern.svg

Airy disc


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The Airy disc

Laser Interference by Petrov Victor. Diffraction of red laser beam by a circular aperture.URL: http://en.wikipedia.org/wiki/File:Laser_Interference.JPG

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The Rayleigh criterion

The first zero of the intensity profile for diffraction from a circular aperture occurs at

.22.1sinD

λθ ≈

This represents the minimum angular separation that two points can be so that they may be separately resolved.

Using the small angle approximation, this becomes

.22.1D

λθ ≈

This is known as the Rayleigh criterion.


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Diffraction limited imaging

Intensity profiles for two resolvable distant point sources.

Merged intensity profiles for unresolvable distant point sources.

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Diffraction from a square aperture

Square diffraction (public domain). A computer simulation of the intensity pattern formed by a laser of 663 nm wavelength incident on a square aperture of 20 by 20 micrometer, visible on a screen placed 1 meter from the aperture. In reality the image measures 30 by 30 cm.URL: http://en.wikipedia.org/wiki/File:Square_diffraction.jpg


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Intensity profile for a double slit

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Multiple slit diffraction


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Multiple slit diffraction (transmission)

Assume incident wavevector on left-hand-side is parallel to z.

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For N slits, the total contribution of the field E

Pis

[ ] .sinsin1

0

2/

2/∑∫

−

=

+

−+−=

N

n

na

Dna

L

PdxkxkRt

R

EE θω

Again, we make use of the earlier approximations for far field diffraction.


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,sin

sin

1

0

21

0

2/

2/

sin

∑∑−

=

−

=

+

−

=

N

n

ni

N

n

na

Dna

ikx

Deik

e

ββ

θα

θ

Focussing on the x-dependent part of the integral and factorising as before, we find

The new factor is a geometric progression with common factor ei2α

∑−

=

=1

0

2 .N

n

ni

NeS

α

where

.sin2

θαka

=

22


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Multiplying SNby ei2α

so

,1

22 ∑=

=N

n

nii

NeeS

αα

( ) ,11 22 αα Nii

NeeS −=−

which gives

( ) .sin

sin1

ααα N

eSNi

N

−=


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Intensity profile for multiple slits

The phase factor may be dropped from this expression. Note also that since

it is useful to incorporate a normalising factor 1/N into this ratio. Hence, the intensity takes the form

,sin

sinlim

0N

N=

→ αα

α

( ) ( ) .sin

sin

sin0

22

=ββ

αα

θN

NII

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Intensity profile for multiple slits

The diffraction condition

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Diffraction condition

Note that the condition for constructive interference is

.sin λθ ma =


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We can re-write this as

.sin2

mka

πθ =

But this is just,mπα =

( ) ( ) .sin

sin

sin0

22

=ββ

αα

θN

NII

which gives the condition for the local maxima of the intensity

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Diffraction – wavelength dependence

.sin λθ ma =

Red (longer wavelength) light is diffracted to a greater extent than

blue (shorter wavelength).

(Yellow arrow is incident light and specular reflection)


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Incident and diffracted wavevectors:

,zk k=

( ).cosˆsinˆ' θθ zxk += k

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Diffraction condition: off-axis incidence

.sin,sinmi

aOBaAO θθ ==

Diffraction gratings

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The grating equation

( ) .sinsin λθθ maim

=+

For off-axis transmission, the diffraction condition is now

This reduces to the case of normal incidence when .0=i

θ

This result may be further generalised by taking the incident angle around to the front of the grating – i.e. making the grating into a reflection grating.


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Reflection gratings

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Reflection gratings

Light strikes the reflection grating at an angle θ

i. For certain angles

θm, the diffraction condition will

be met:

The path lengths of rays from the incident

wavefront via the successive rulings of the

grating and leaving at the same angle must

differ only be integral multiples of the

wavelength λ.


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Derivation of the reflection grating equation

Incident wavefront AC

Path from A to wavefront BD

.sinm

aAB θ=

Path from C to wavefront BD

.sini

aCD θ=

( ).sinsinim

aCDAB θθ −=−Path difference

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The reflection grating equation

The diffraction condition for a reflection grating may then be expressed mathematically as

( ).sinsinim

am θθλ −=

This is known as the reflection grating equation.


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Dispersion

The dispersion of a grating is defined as

.λθ

θd

dD

m=

Differentiating the grating equations, we have

.cosm

m

ad

dm θ

θλ

=To be solved in the

Problem Set!

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Dispersion

Hence,

.cos

ma

mD

θθ =

Thus, we have greater dispersion for

• higher order m• smaller slit spacing a

To be solved in the

Problem Set!


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Number of orders

Recall that

So the highest order m is the largest integral value of

( ) .sinsin λθθ maim

=±

( ).sinsinim

aθθ

λ±

.2

max λa

m <

Hence

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Resolving power

The resolving power of a grating is defined as

,λλ∆

=R

where, via Rayleigh’s criterion, ∆λ is the minimum resolvable wavelength between the peaks of two wavelengths with midpoint λ.


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Common example of a grating

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Monochromators


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The Czerny-Turner monochromator

collimator

diffraction grating

camera

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• Entrance slit

• Collimator (concave mirror)

• Diffraction grating

• Camera (concave mirror)

• Exit slit

Czerny-Turner monochromator


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Optical spectrum analyser (OSA)

Optical spectrum analyser (OSA) based on diffraction gratings

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Spectroscopy – blackbody spectra


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Spectroscopy – emission and absorption lines

samplelight in

emission

light out - absorption

bright lines

dark lines

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Spectroscopy – chemical analysis

Spectra provides `chemical fingerprint’

Emission spectrum of iron

Diffraction around objects

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diffraction at edge of object


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similar to diffraction through a slit

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‘large’ object (compared to wavelength)


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‘small’ object (compared to wavelength)

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Other examples of diffraction


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Diffraction around a razor blade

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X-ray diffraction (non-optical)

(See PY3105)


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Related phenomena:

Thin-Film Interference

(See Chapter 9, PY3101)

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PY3101 Optics - Physics | University College CorkDepartment of Physics PY3101 Optics Diffraction...

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