What is diffraction?
The process where light spreads when it passes through a narrow aperture.
When is light diffracted?
If the physical size of the aperture is comparable to, or smaller than, the wavelength of the light
What is Fraunhofer Diffraction?
Waves incident on and leaving the aperture can be considered plane
What is Fresnel Diffraction?
Waves incident on and leaving the aperture can NOT be considered plane
What is the geometry for observing Fraunhofer diffraction?
- 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
What are the steps of calculating Fraunhofer diffraction produced by an (1D) aperture?
- 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)
How do we get a aperture that has a variable value?
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
What does the diffraction depend on?
- Depends on the size of the aperture.
- aperture size»_space; wavelength
- light travels in a straight line with no diffraction as a result we get geometrical optics and shadows etc
What is the width of the central peak of the diffraction pattern by a single slit?
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
What are the steps of calculating Fraunhofer diffraction produced by an (2D) aperture?
- 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
What are the equations for the diffraction by a rectangular aperture?
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
What are the equations for the diffraction by a circular aperture?
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
What are the effects of diffraction on spatial resolution of a telescope?
Diffraction as light passes through a telescope may prevent closely spaced objects from being separately resolved.
What are the equations for the diffraction by two very narrow slits?
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
What are the equations for the diffraction by two slits of non-zero width?
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
What is the consequence of diffraction by two slits of non-zerowidth?
Missing bright fridges in the diffraction pattern (missing orders)
What are the equations for diffraction by an array of N identical slits?
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
What is the diffraction equation and what does it tell us?
𝑑 sin𝜃=𝑛𝜆
(for large N)
Tells us the angles where intense light is diffracted.
What is the equation for the angular width of the main maxima of the diffracted light?
Δ𝜃=𝜆 2𝜋/(𝑑𝜋 cos𝜃 )𝑁
What is the equation for the number of diffraction orders produced by a diffraction grating?
nmax= d/𝜆
n must be integer it must take the next lowest whole number
What is the Rayleigh criterion for spectral resolution?
When the maxima of one wavelength coincides with the first minima of the second wavelength
What is the equation for the spectral resolving power?
𝜆/Δ𝜆
𝜆 - average wavelength
Δ𝜆 - the difference between two wavelength
What is a diffraction grating and why is it used?
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
What is the equation for the spectral resolving power for a diffraction grating?
𝜆/Δ𝜆=𝑛𝑁 n - diffraction order N - slit number 𝜆 - average wavelength Δ𝜆 - the difference between two wavelength
