Diffraction

Question 1

What is diffraction?

Answer 1

The process where light spreads when it passes through a narrow aperture.

Question 2

When is light diffracted?

Answer 2

If the physical size of the aperture is comparable to, or smaller than, the wavelength of the light

Question 3

What is Fraunhofer Diffraction?

Answer 3

Waves incident on and leaving the aperture can be considered plane

Question 4

What is Fresnel Diffraction?

Answer 4

Waves incident on and leaving the aperture can NOT be considered plane

Question 5

What is the geometry for observing Fraunhofer diffraction?

Answer 5

• Large distances between S(source) and screen and also the screen and L(second screen)
  ^ Too big of a system over come by:
  OR
• Two lenses to produce incoming and outgoing plane waves ( diffraction pattern is produced in focal plane of second lens) – at a distance f from L

Question 6

What are the steps of calculating Fraunhofer diffraction produced by an (1D) aperture?

Answer 6

• Write f(x) (describes the transmission of light through the screen)
• Calculate the Fourier transform of f(x) to give F(kx)
• Calculate |F(kx)|^2
  Substitute kx =2πsintheta/lambda
  Replace sintheta with x/f

Fraunhofer diffraction by a single long square slit:
𝐼(𝑥)=𝐼_0 sinc ^2 (𝜋𝑤x/𝑓𝜆)
Fraunhofer diffraction by a single long triangle slit:
𝐼(𝑥)=𝐼_0 sinc ^4 (𝜋𝑤x/𝑓𝜆)

Question 7

How do we get a aperture that has a variable value?

Answer 7

Systems with transmission functions intermediate between one and zero
For example using glass with filter, if it is very thin it will partial transmit light. Used in photography

Question 8

What does the diffraction depend on?

Answer 8

• Depends on the size of the aperture.
• aperture size&raquo_space; wavelength
• light travels in a straight line with no diffraction as a result we get geometrical optics and shadows etc

Question 9

What is the width of the central peak of the diffraction pattern by a single slit?

Answer 9

Δx=2sλ/w ( s or f )
Example 15

Δx -the separation between fringes 
s - the separation between screens 
f - distance to screen L when a lens is used
λ - wavelength
w - the slit width

Question 10

What are the steps of calculating Fraunhofer diffraction produced by an (2D) aperture?

Answer 10

• Write f(x) (describes the transmission of light through the screen)
• Calculate the Fourier transform of f(x) to give F(kx)= 𝐹(𝑘_𝑥,𝑘_𝑦)=1/√2𝜋 1/√2𝜋 ∫(−∞)^(+∞)∫(−∞)^(+∞)𝑓(𝑥,𝑦)𝑒^(−𝑖𝑘_𝑥 𝑥) 𝑒^(−𝑖𝑘_𝑦 𝑦) d𝑥d𝑦 (In 2D)
• Calculate |F(kx)|^2
  Substitute kx =2pisintheta/lambda
  Replace sintheta with x/f

Question 11

What are the equations for the diffraction by a rectangular aperture?

Answer 11

Δx=2sλ/wx
Δy=2sλ/wy
( s turns into f for a lens)
Δx - the spot size (the diameter of the central maxima of diffraction pattern) 
s - the separation between screens 
λ - wavelength
w - the slit width/size of the aperture

Question 12

What are the equations for the diffraction by a circular aperture?

Answer 12

Δx =~2.44λs/D
( s turns into f for a lens)
Δx - the spot size (the diameter of the central maxima of diffraction pattern)
s - the separation between screens
f - focal length/distance to screen L when a lens is used
λ - wavelength
w - the slit width

Question 13

What are the effects of diffraction on spatial resolution of a telescope?

Answer 13

Diffraction as light passes through a telescope may prevent closely spaced objects from being separately resolved.

Question 14

What are the equations for the diffraction by two very narrow slits?

Answer 14

Fridge spacing:
Δx =λs/d (s or f)
s - the separation between screens 
f - distance to screen L when a lens is used
λ - wavelength
d - the separation between slits 
Δx -the separation between fringes

Question 15

What are the equations for the diffraction by two slits of non-zero width?

Answer 15

𝐼(𝑥)=𝐼_0 sinc ^2 (𝜋𝑤/𝑓𝜆 𝑥) cos^2⁡(𝜋𝑑/𝜆𝑓 𝑥)
First term - Diffraction by a single slit
Second term - Diffraction by two
very narrow slits

IGNORING THE FIRST TERM
Intense maxima occur when denominator term is zero which gives Δx =λs/d, we maximise d by choosing the one at the very end d = N-1/2 * centre of slit spacing

Question 16

What is the consequence of diffraction by two slits of non-zerowidth?

Answer 16

Missing bright fridges in the diffraction pattern (missing orders)

Question 17

What are the equations for diffraction by an array of N identical slits?

Answer 17

𝐼(𝑥)=𝐼_0sinc^2 (𝜋𝑤x/𝜆𝑓) (sin⁡(𝑁𝑑𝜋x/𝜆𝑓)/sin⁡(𝑑𝜋x/𝜆𝑓 )^2

First Term - Diffraction by a single slit
Second Term - Diffraction by a N very narrow slits

IGNORING THE FIRST TERM
Intense maxima occur when denominator term is zero this gives the diffraction equation

Weaker maxima occur when numerator equals one

Question 18

What is the diffraction equation and what does it tell us?

Answer 18

𝑑 sin⁡𝜃=𝑛𝜆
(for large N)
Tells us the angles where intense light is diffracted.

Question 19

What is the equation for the angular width of the main maxima of the diffracted light?

Answer 19

Δ𝜃=𝜆 2𝜋/(𝑑𝜋 cos⁡𝜃 )𝑁

Question 20

What is the equation for the number of diffraction orders produced by a diffraction grating?

Answer 20

nmax= d/𝜆

n must be integer it must take the next lowest whole number

Question 21

What is the Rayleigh criterion for spectral resolution?

Answer 21

When the maxima of one wavelength coincides with the first minima of the second wavelength

Question 22

What is the equation for the spectral resolving power?

Answer 22

𝜆/Δ𝜆
𝜆 - average wavelength
Δ𝜆 - the difference between two wavelength

Question 23

What is a diffraction grating and why is it used?

Answer 23

A structure consisting of a large number of slits (e.g. N>100)

used to split light consisting of many wavelengths into the wavelength components

Question 24

What is the equation for the spectral resolving power for a diffraction grating?

Answer 24

𝜆/Δ𝜆=𝑛𝑁
n - diffraction order
N - slit number
𝜆 - average wavelength
Δ𝜆 - the difference between two wavelength

Question 25

What are the consequences of diffraction?

Answer 25

When light of wavelength passes through the aperture of a telescope of diameter D and if the focal length is f then the angular and spatial widths of the maximum of the resultant diffraction pattern are

Δ𝜃≈2.44 𝜆/𝐷, Δ𝑥≈2.44 𝜆𝑓/𝐷

If light from two closely spaced objects enters a telescope the spread in their images that results from diffraction may prevent them from being separated – spatially resolved

There will be a minimum angular separation that two objects can have such that the telescopes shows them as being separate – this is the angular resolution of the system

Question 26

What is the equation for the size of an image if diffraction is IGNORED and not ignored?

Answer 26

Δ𝑥 = αf
α - angular power
f - focal length

Δ𝑥≈2.44 𝜆𝑓/𝐷
𝜆 - wavelength
𝑓 - focal length
D - diameter

If 2nd effect&raquo_space; 1st so just use 2nd effect

Question 27

What is the angular resolution of an optical system/resolving power of a telescope?

Answer 27

Δ α = 1.22𝜆/𝐷

Δ α - resolving power
D - input aperture diameter
𝜆 - imaging light of wavelength

Question 28

What is the equation for the maximum distance an optical system can resolve light?

Answer 28

Δ α = x/d

x- separation
d - min distance
α - angular resolving power (minimum angular separation between two points)