pc20312 wave optics section 4: diffraction. huygens-fresnel principle i image from wikipedia...

$: PC20312 Wave Optics Section 4: Diffraction. Huygens-Fresnel Principle I Image from Wikipedia Augustin-Jean Fresnel 1788-1827 “Every unobstructed point$
PC20312 Wave Optics

Section 4:

Diffraction

Huygens-Fresnel Principle I

Image from Wikipedia

Augustin-Jean Fresnel 1788-1827

“Every unobstructed point of a wavefront… serves as a source of spherical secondary wavelets … The amplitude of the optical field at any point beyond is the superposition of all these wavelets ...”

• Fresnel combined ideas of Huygens’ wavelets & interference

• Postulated in 1818:

Hecht, p444

Huygens-Fresnel Principle II

Gustav R. Kirchhoff

1824-1887

• Fresnel’s postulate (1818) predates Maxwell’s equations (1861)

• Formally derived from the scalar wave equation by Kirchoff in 1882

• Worked with Schuster for year at the University of Heidelberg


Huygens-Fresnel Principle III

Total area, A

P

dA1

dA3dA2

Optical field at P depends on the superposition of contributions from each elemental area dA of

the total area A

Huygens-Fresnel Principle IV

Divide an aperture into elemental areas each of which is a source of a spherical wavelet


http://www.acoustics.salford.ac.uk/feschools/waves/diffract3.htm

The Huygens-Fresnel Integral

Source, S Observation point, P

Q

Spherical wavefront

s r

s0 R

Fraunhofer diffraction

Joseph von Fraunhofer

1787-1826

The case of small, linear phase variation, i.e.:

• r R + r ,

• r << R

• r x,y

Satisfied when s,r >> d

Hence, “Far-field diffraction”


y

x

aperture

d

Far-field diffraction

S

s0

• s0 >> d

• wavefront plane at aperture

• s s0

P

d D

• R >>d

• const. set K() 1

• D >> d

R

Analysis of Fraunhofer diffraction

Source, S

Observation point, P(X,Y)

Q(x,y)

s

r

R

s0 Z

Aperture, A(x,y)

Single slit diffractiony

xa/2-a/2


Rectangular aperturey

xa/2-a/2

b/2

-b/2


Circular aperture Iy

xa

u


Airy disc

Airy rings

The Airy Pattern

Circular aperture II

-10 -5 0 5 100

0.2

0.4

0.6

0.8

1

No

rma

lise

d I

nte

nsi

ty

ka

kaθD=3.83I=0.0175I(0)

The diffraction limit

f

f Image from Google Images

If there was no diffraction:

• parallel rays focused to a point

• images would be perfectly sharp

BUT, diffraction from instrumental apertures :

• produce rays at a range of angles

• which are focused at different points

• image is thus smeared out.

Even for a perfect optical system, diffraction limits

resolution.

Radius of the Airy disc

f

D

Radius, RA= fD = 1.22f/d

Fraunhofer diffraction patterns also formed in focal plane of a lens¶

¶ e.g. see ‘Modern Optics’ by R Guenther

Appendix 10-A

Two finite slits

E2(X)

E1(X)

E1(X)E2(X)

d

R

x

X

aa

Image courtesy of A Pedlar

Point spread function

Images courtesy of A Pedlar & from Wikipedia

The diffraction grating

David Rittenhouse 1732-1796

• Rittenhouse 1785:

• fine threads between screws – 100 threads/inch

• Fraunhofer 1821:

• thin wires

• Henry Augustus Rowland:

• curved gratings

• spectrocopy

• Henry Joseph Grayson 1899:

• developed precise ‘ruling engine’

• 120,000 lines/inch

Henry Augustus Rowland 1848-1901

A periodic structure designed to diffract light

Images from Wikipedia

Grating structure

Ruled grating

Blazed grating – enhances diffraction in one direction

Gratings:

• central to modern spectrometers

• reflection or transmission

• amplitude or phase

Phase grating

0 5 10 15Distance

Ref

ract

ive

inde

x

Analysis of diffraction from gratings

d

d

1 2

Path length difference for incident rays: Path length difference for diffracted rays:

d

1

2

Modern gratings

Reflection gratings

Transmission gratings

CDs / DVDsImages from Wikipedia

Gratings in nature

Nacre

Peacock feathers

Butterfly wings

Images from Wikipedia

Grating based spectrometers

The Czerny-Turner monochromator.

• A – input light

• B – entrance slit

• C – collimating mirror

• D – diffraction grating

• E – focusing mirror

• F – exit slit

• G – output light


General diffraction (again)

Source, S Observation point, P

Q

Spherical wavefront

s r

s0 R

Half-period zones

s

S

P

rm+1

rm

rm

rm+1

S P

Area of the mth zone

S

ds

sd

ssin

P

ssin

ssind

S

s

P

rm

s+R

Zone plates

Arago’s spot

François Jean Dominique Arago (1783-1856)

Siméon Denis Poisson (1781 -1840)

http://demo.physics.uiuc.edu/LectDemo/scripts/demo_descript.idc?DemoID=749

Merde !

Fresnel diffraction from straight edges

y

x

Q(x,y)

S

s

P

r

Q(x,y)

s0 R

pc20312 wave optics section 4: diffraction. huygens-fresnel principle i image from wikipedia...

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