Diffraction by N-slits

/ 2 / 2

/ 2 / 2

2 / 2 ( 1) / 2

2 / 2 ( 1) / 2

( ) ( )

( ) ........ ( )

( ) sin[ ( sin )]

b a b

b a b

a b N a b

a b N a b

E C F z dz F z dz

F z dz F z dz

F z t k R z

Optical disturbance due to N slits

Contribution of jth slit

sinsin( 2 )

Provided

where, sin and sin2 2

j

j

E bC t kR j

kb ka

Contribution of N slits

1

0

1( ) 2

0

sinIm ( )

sin sinsin[ ( 1) ]

N

j

Ni t kR i j

j

E E

bc e e

NbC t kR N

2 2

0

20

sin sin( ) ( ) ( )*

sin

(0)

NI E E I

I N I

Irradiance due to N-slits

I0 = Irradiance by single slit at =0

For principal maxima

sin for 0, , 2

sin

NN

or

sin 0, 1, 2....ma m m

For minima

sin0

sin2 ( 1) ( 1)

for , ,....., , ,..

N

N N

N N N N

Between consecutive principal maxima, there will be N-1 minima

Subsidiary maxima

3 5 for , ,.....

2 2N N

(N-2) subsidiary maxima between consecutive principle maxima

For large N, irradiance of first subsidiary maxima

2 2sin 2

(0)3

I I

Diffraction from multiple slits

Slit Diffraction

Pattern Pattern

Normal incidence

Transmissiongrating

Oblique incidence

sin sin ma i m

a

For two wavelengths

Dispersive power of grating

Width of principal maxima

cosw

mNa

sD

cos m sa m

cos m

mD

a

Barely resolved

It

s

w

Chromatic resolving power of a grating

ws

cos m

mNa

mNPRCbare

..

Chromatic resolving power of a prism

d

dBPRC

bare

..

Reflection grating md m sin

Diffraction grating

The nominal track separation on a CD is 1.6 micrometers, corresponding to about 625 tracks per millimeter. This is in the range of ordinary laboratory diffraction gratings. For red light of wavelength 600 nm, this would give a first order diffraction maximum at about 22° .

15 15

0 nm

450 nm

Atomic Force Microscope Image of CD

Bragg’s law X-ray diffraction from crystals:2d Sin θ = n λ

2( ) (4 cos / )cosn

n

x nI x J h i

z i

Phase grating

Delta functionBessel function

ProblemConsider a opaque screen with 5 equally spaced narrow slits (spacing between them is d) and with monochromatic plane wave (wavelength ) incident normally.Draw a sketch of the transmitted intensity vs. angle to the normal for = 0 to = 1/5 radian. Take Sin = over this range and assume d/ =10. What is the ratio of least intense to the most intense peak?What is the angular distance of the first intense peak away from = 0.