Fraunhofer DiffractionFraunhofer Diffraction

Last Lecture• Dichroic Materials• Polarization by Scattering• Polarization by Reflection from Dielectric Surfaces• Birefringent Materials• Double Refraction• The Pockel’s Cell

This Lecture• Fraunhofer versus Fresnel Diffraction• Diffraction from a Single Slit• Beam Spreading• Rectangular and Circular Apertures

Optical Diffraction Optical Diffraction

• Diffraction is any deviation from geometric optics that results from the obstruction of a light wave, such as sending a laser beam through an aperture to reduce the beam size. Diffraction results from the interaction of light waves with the edges of objects.

• The edges of optical images are blurred by diffraction, and this represents a fundamental limitation on the resolution of an optical imaging system.

• There is no physical difference between the phenomena of interference and diffraction, both result from the superposition of light waves. Diffraction results from the superposition of many light waves, interference results from the interference of a few light waves.

Optical Diffraction Optical Diffraction

Hecht, Optics, Chapter 10

Fraunhofer versus Fresnel Diffraction Fraunhofer versus Fresnel Diffraction

• The passage of light through an aperture or slit and the resulting diffraction patterns can be analyzed using either Fraunhofer or Fresnel diffraction theory. In Fraunhofer (far-field) diffraction theory the source is far enough from the aperture that the wavefronts are planar at the aperture, and the image plane is far enough from the aperture that the wavefronts are planar at the image plane.

• If the curvature of the optical waves must be taken into account at the aperture or image plane, then we must use Fresnel (near-field) diffraction theory.

• The Huygens-Fresnel principle is used in diffraction theory, in that every point of a given wavefront of light can be considered as a source of secondary wavelets. To analyze two-slit interference, we assumed that the slits were point sources. To analyze diffraction, we need to consider the generation of wavelets at different spatial positions within the slit.

Fraunhofer versus Fresnel Diffraction Fraunhofer versus Fresnel Diffraction

• We can move to the Fraunhofer regime by placing lenses on each side of the aperture. Lens 1 is place a focal length away from the point source so that the wavefronts are planar at the aperture. The observation screen is in the focal plane of lens 2 so that the diffraction pattern is imaged at infinity.

Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit

• Consider the geometry shown below. Assume that the slit is very long in the direction perpendicular to the page so that we can neglect diffraction effects in the perpendicular direction.

7

Fraunhofer vs. Fresnel diffractionFraunhofer vs. Fresnel diffraction

• In Fraunhofer diffraction, both incident and diffracted waves may be considered to be plane (i.e. both S and P are a large distance away)

• If either S or P are close enough that wavefront curvature is not negligible, then we have Fresnel diffraction

P

S

Hecht 10.2Hecht 10.2 Hecht 10.3Hecht 10.3

8

Fraunhofer Vs. Fresnel DiffractionFraunhofer Vs. Fresnel Diffraction

2

2222

22222222

'

11

2

1

'

'

2

11

'

'

2

11'

2

11

'

'

2

11'

'''

ddd

h

d

h

d

hd

d

hd

d

hd

d

hd

hdhdhdhd

Now calculate variation in (r+r’) in going from one side of aperture to the other. Call it

9

Fraunhofer diffraction limitFraunhofer diffraction limit

• If aperture is a square - X • The same relation holds in azimuthal plane

and 2 ~ measure of the area of the aperture

• Then we have the Fraunhofer diffraction if,

apertureofaread

or

d

,

2

Fraunhofer or far field limit

10

Fraunhofer, Fresnel limitsFraunhofer, Fresnel limits

• The near field, or Fresnel, limit is

2

d

Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit

0

0

0

0

exp

0.

P

P

The contribution to the electric field amplitude

at point P due to the wavelet emanating from

the element ds in the slit is given by

dEdE i kr t

r

Let r r for the source element ds at s

Then for any element

dEdE

r

0

0

exp

, .

, , .

sin

L L

i k r t

We can neglect the path difference in the amplitude term but not in the phase term

We let dE E ds where E is the electric field amplitude assumed uniform over the width of the slit

The path difference s

/ 2

0 0 / 20 0

/ 2

00 / 2

.

exp sin exp exp sin

exp sinexp

sin

bL L

P P b

b

LP

b

Substituting we obtain

E ds EdE i k r s t E i kr t i k s ds

r r

i k sEIntegrating we obtain E i kr t

r i k

Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit

00

00

00

exp expexp

sin

1sin

2

exp exp exp2

exp2

LP

LP

L

Evaluating with the integral limits we obtain

i iEE i kr t

r i k

where

k b

Rearranging we obtain

E bE i kr t i i

r i

E bi kr t

r i

00

2 2 2* 2

0 0 0 02 20

sin2 sin exp

1 1 sin sinsinc

2 2

L

LP P

E bi i kr t

r

The irradiance at point P is given by

E bI = c E E c I I

r

Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit

2 2 2* 2

0 0 0 02 20

0 0

1 1 sin sinsinc

2 2

sinsinc 1 0, lim sinc lim 1

1sin 0, sin 1, 2,

2

LP P

The irradiance at point P is given by

E bI = c E E c I I

r

The function is for

The zeroes of irradiance occur when or when k b m m

Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit

,

sin , 2 / ,

1 2

2

In terms of the length y on the observation screen

y f and in terms of wavelength k

we can write

y b yb

f f

Zeroes in the irradiance pattern will occur when

b y m fm y

f b

The maximum in the irradiance pattern is at β = 0

2 2

sin cos sin cos sin0

sintan

cos

.

Secondary maxima are found from

d

d

Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit

Note: x- and y-axes switched in book, Figs. 16-5a (here) and Fig. 16-1 do not match.

Beam SpreadingBeam Spreading

sin

min

1 2

The angular width of the central

maximum is defined as the angular

separation Δθ between the first minima on

either side of the central maximum,

y

f

The first ima in the irradiance pattern

will occur when

fm fy Δθ

b b b

The

2

width W of the diffraction pattern thus

increases linearly with distance from the slit,

in the regions far from the slit where Fraunhofer

diffraction applies

LW = L Δθ

b

Rectangular AperturesRectangular Apertures

When the length a and width b of the

rectangular aperture are comparable,

a diffraction pattern is observed in

both the x - and y - dimensions, governed

in each dimension by the formula we

have already developed. The irradiance

pat

2 20 sinc sinc

1sin

2

tern is

I I

where

k a

Zeroes in the irradiance pattern are observed

when

m f m fy or x

b a

x

y

Square AperturesSquare Apertures

Fraunhofer Diffraction from General Apertures

Fraunhofer Diffraction from General Apertures

1/ 22 22

1/ 22 2 2

expAP

For the general aperture

EdE i t kr dA

r

where

dA dy dz

r X Y y Z z

R X Y Z

Fraunhofer Diffraction from General Apertures

Fraunhofer Diffraction from General Apertures

1/ 22 2

2 2

1/ 2

2

21

,

21 1

2 2

2

We can combine the relations for r and R to obtain

y z Yy Zzr R

R R

In the far field R is very large compared to the aperture dimensions, and

y + zwecan neglect the term and wecan write

R

Yy Zz Yr R R

R

2

exp exp

exp exp

AP

aperture

A

aperture

y Zz Yy ZzR

R R

Therefore, the total electric field at point P is given by

EE i t kR ik Yy Zz dA

R

Ei t kR ik Yy Zz dy dz

R

Fraunhofer Diffraction from Circular AperturesFraunhofer Diffraction from Circular Apertures

.

cos sin

cos sin

sin sinexp exp

P

AP

Now we specialize to a circular aperture of radius a At this point we switch to cylindrical coordinates

z y

Z q Y q

dA d d

Substituting into our expression for E we obtain

ik qEE i t kR

R

2

0 0

2

0 0

cos cos

exp exp cos

a

aA

qd d

R

E i k qi t kR d d

R R

Fraunhofer Diffraction from Circular AperturesFraunhofer Diffraction from Circular Apertures

2

0 0

2

0

0

.

exp exp cos

exp cos .

1( )

2

aA

P

Because of symmetry our solution will be the same for any angle Choosing Φ = 0, we obtain

E i k qE i t kR d d

R R

k qBut i d is in the form of a Bessel function

R

J u

2

0

00

2

0

exp cos .

exp 2

,

( ) exp cos2

v

v

aA

P

m v

m v

i u v dv is a Bessel function of the order zero

We can write

E k qE i t kR J d

R R

In general a Bessel function of the order m is given by

iJ u i mv u v dv

Fraunhofer Diffraction from Circular AperturesFraunhofer Diffraction from Circular Apertures

1

1

00

,

exp 2

m mm m

u

00

AP

A useful recurrence relation for Bessel functions is

du J u u J u

du

When m = 1, we can integrate the expression to find

J u du u J u

k q RDefine w then d dw gives us

R kq

E k qE i t kR J d

R R

2/

00

21

22* 2

0 1

exp 2

exp 2

1

2

a w kaq RA

AP

AP P

E Ri t kR J w wdw

R kq

From the recurrence relation we obtain

E R kaqE i t kR a J

R kaq R

The irradiance is given by

E R kaqI c E E A J

R kaq R

2

Fraunhofer Diffraction from Circular AperturesFraunhofer Diffraction from Circular Apertures

2 2

1 1

1

0

sin / ,

2 2 sin0 0

sin

1lim

2u

Assuming that R is essentially constant over the observation screen and recognizing that

q R we can write the irradiance as

J kaq R J k aI I I

kaq R k a

where we have used the

J urelation that

u

0

0.

recognizing that u when

q

Fraunhofer Diffraction from Circular Apertures: Bessel Functions

Fraunhofer Diffraction from Circular Apertures: Bessel Functions

Fraunhofer Diffraction from

Circular Apertures: The Airy Pattern

Fraunhofer Diffraction from

Circular Apertures: The Airy Pattern

min

min

2sin 3.83

2

1.22

2 .

First minimum in the Airy pattern is at

Dk a k a

D

where D a

Circular AperturesCircular Apertures