Array theorem and convolutions

Derive array theorem from convolution theorem

( ', ') 2 x xF g h x y g h

individualaperture1

x n y nN

ik x y

array x y x yn

E z e E z

, , * , ,

Review: Sketch the diffraction pattern of this rectangular slit

A) I got it mostly rightB) I got it mostly wrong but I tried

The E-field at the screen due to one slit is (centered at the origin)

A) I got it mostly rightB) I got it mostly wrong but I tried

sinc2 2o x y

ka kbE

Find and sketch the E-field at the screen due to due slits centered at y = ±h/2.

Diffraction of the light from a distant flashlight viewed through a black umbrella fabric.

2-slit Fraun. diffraction pattern

How many minor peaks are between the major peaks?

N-slits diffraction features

http://optics.byu.edu/animation/fraunhof.mov

N-slits, grooves,mirror-strips diffraction grating

2 2

1 2 2

2

2

x x

x n

x x

kh khi N i N

xNi x

kh khi in

x

khN

e ee

khe e

'

sin

sin

2

2peak

2 2

sinsinc

sin

hN

aI I

hN

major peaks occur when:

minor peaks occur when:

N-slits, grooves,mirror-strips diffraction grating

2

2peak

2 2

sinsinc

sin

hN

aI I

hN

u m

Nu

u

sinlim

sinRatio of major to minor peak in I:a) N b) N2 b) N1/2

Width of major peak decreases as what power of N?a) 1/N b) 1/N2 b) 1/N1/2

Diffraction grating orders

Diffraction grating orders

Reflective gratings are “blazed”

Reflective gratings most common: lines cut in glass, replicated onto plastic blanks, coated with metal (aluminum)

Typical grating for visible to UV light: 1000-2000 grooves/mm, 10 cm wide105

lines.

Modern methods

Blazed gratings put more light into a nonzero order

Resolving power of a grating

2

2peak

2 2

sinsinc

sin

hN

aI I

hN

peakRP mN

How much must l change so peak moves by half its width?

Spectrometer

Put slits at f from mirrors. Grating in center