experiment 7 - diffraction limited resolution in optical systems and the eye

Question 1

what are we investigating in this experiment ?

Answer 1

• how diffraction and in particular with how the airy pattern limits the final spatial detail a lens can resolve

Question 2

what is spatial resolution limited by ?

Answer 2

• limited by diffraction , higher order aberrations and scattered light

Question 3

what does every circular aperture produce ?

Answer 3

• every circular aperture produces a diffraction pattern

Question 4

when does diffraction affect size of the airy diskc?

Answer 4

• diffraction affects the size of the airy disc produced by the lens when imaging a very small dot object
• it is expected that diffraction takes place only at the edge of the aperture

Question 5

explain the symmetry of airy disc ?

Answer 5

• the airy disc has radial symmetry and it consists of a centre disc of radius , p , surrounded by a number of progressively larger annuli
• surrounding annuli are much dimmer than the centre disc and appear both dark and bright

Question 6

what kind of aperture are the pupil and lens ?

Answer 6

• the pupil and the lens diaphragms are also circular aperture

Question 7

does a rectangular or slit aperture produce same diffraction pattern as a circular aperture ?

Answer 7

• a rectangular or slit aperture produce a different diffraction pattern
• no longer airy pattern

Question 8

what bit of the aperture is involved in diffraction ?

Answer 8

• only the edge of aperture stop is involved in diffraction

Question 9

what is the amount of diffracted light proportional to ?

Answer 9

• since only the edge of aperture is involved in diffraction , the amount of diffracted light is proportional to the length of its circumference
  2πr

Question 10

what does the area of aperture increase with ?

Answer 10

the area of aperture increases with the square of the radius πr^2

Question 11

what is the fraction captured by the lens that is diffracted proportional to ?

Answer 11

1/r
r=radius of aperture
- this ratio changes as you change aperture size

• this is because circumference is 2πr ( light that is diffracted is proportional to 2πr ) and radius is πr^2 -( light that is un diffracted is proportional to πr^2)
• we take ratio of circumference to area we get 1/r

Question 12

what happens when pupil diameter is 3mm?

Answer 12

• the eye is free from on-axis optical aberrations

- when this happens the spatial resolution of the eye is limited by only diffraction

Question 13

what is the hypothesis for this experiment ?

Answer 13

• the fraction of the light that is is diffracted when passing through the lens is proportional to the ratio of the circumference to the area of the aperture stop
• fraction of diffracted light = 2πr / πr^2
• ratio determines the size of airy disc which is diffraction pattern
  p ∝ 1/r

Question 14

what is is the equation for fraction of diffracted light ?

Answer 14

• fraction of diffracted light = 2πr / πr^2
• ratio determines the size of airy disc which is diffraction pattern
  p ∝ 1/r
• the radius (p) of the centre lobe in the airy pattern

Question 15

how test to test our hypothesis ?

Answer 15

• use diffraction limited object ( i.e. vey small light emitting dot ) and produce an image for this point-like object for each aperture size investigated
• the microscope is viewing the airy disc , and look at image magnified with microscope objective
• what we are measuring is the diameter of the central lobe which is the same as measuring the first dark ring which is the twice the radius as the first dark ring
• we need to vary the aperture size for smallest to largest
• for each aperture size we need to measure size of airy disc

Question 16

how do we measure aperture size ?

Answer 16

• we need to measure the size of the radius of the first dark ring ( as a measure of the airy disc) . the travelling microscope must have a fine micrometer scale to measure the radius of the first dark ring for every pupil size
• record radius of aperture (r) and the corresponding radius of the airy disc p

Question 17

what do we see when we plot graph for aperture diameter ( x-axis ) and ratio ( aperture circumference / area ) ( y-axis )?

Answer 17

1- aperture diameter ( x-axis ) and ratio ( aperture circumference / area ) ( y-axis )

• for large pupil size the amount of diffracted light is only a small fraction of total
• for a small aperture size , the amount of diffracted light is larger

Question 18

what do we see when we plot airy disc radius um ( y-axis ) against aperture diameter ( x-axis ) ?

Answer 18

• for large pupil size the amount of diffracted light is only a small fraction of total
• for a small aperture size , the amount of diffracted light is larger

Question 19

what doe we get we we plot radius of airy pattern ( y-axis ) and the ratio of circumference to pupil area ( x-axis ) ?

Answer 19

• we get a straight line

- radius of airy disc is directly proportional to the ratio of circumference to pupil area

Question 20

what do our results from the graph tell us ?

Answer 20

• the aperture size controls the fraction of light diffracted by the lens which in turn determines the size of the radius of the first dark ring in the airy pattern
• size of airy disc relates to aperture diameter and that relates to the amount of diffraction produced of lens

Question 21

the radius of first disc in diffraction pattern is proportional to ?

Answer 21

p ∝ 0.61λ / n’u’

• directly proportional to the wave length of light
• inverse proportional to refractive index n’ and and to the angle u’

Question 22

the radius of first disc ( airy pattern ) in diffraction pattern is proportional to ?

Answer 22

p ∝ 0.61λ / n’u’

• n’ = refractive index
• u’ = the maximum convergence angle of the rays
• directly proportional to the wave length of light
• inverse proportion al to refractive index n’ and and to the angle u’

Question 23

why is the aperture size important ?

Answer 23

• the bigger the lens the bigger the angle u’

Question 24

what does Rayleigh criterion suggest ?

Answer 24

Images of adjacent points in the object that can still be resolved
clearly(at the Rayleigh limit) when their separation equals the radius of the first
dark ring in the partially overlapping adjacent images.

Question 25

how do we predict p ( airy disk ) as function of pupil size without having to measure it ?

Answer 25

Use the limit of resolution based on the Rayleigh criterion to calculate the smallest angle, bo , the eye can resolve and express this angle in minutes of arc.

Question 26

what Rayleigh criterion definition ?

Answer 26

The minimum separation between two adjacent object points that can be resolved must produce an image separation equal to, r, (the radius of the first dark ring in the Airy disc)

Question 27

how to convert from spatial resolution b0 min of arc which is resolving power of the optics of eye to snellen notation ?

Answer 27

VA = 6/(6* bo) - if object field angle bo is 1 then VA will be 6/6 - if object field angle bo is 2 then VA will be 6/12 - if object field angle bo is 3 then VA will be 6/18

Question 28

what do we observe when plotting visual acuity (min arc ) (y-axis ) against pupil diameter ( mm) ( x-axis )?

Answer 28

The graph shows the limits imposed on the best spatial resolution (min arc) that can be achieved with the optics of the eye as a function of pupil size. The estimates are based on the Rayleigh criterion and assume an eye free of optical aberrations. The numbers on the right of the figure show the corresponding Snellen acuities.

Question 29

what is the typical pupil size under photopic ambient light ?

Answer 29

- 3.5 to 4.5 mm we expect to get 6/3 acuity according to the graph

Question 30

what happens once pupil diameter exceeds 3.5 mm ?

Answer 30

- once you exceed 3.5 the eye has significant amount of aberration - no longer affected by diffraction but with residual aberration in eye

Question 31

what will happen if we reduce light levels so pupil size is 7 to 7.5 mm ?

Answer 31

- we end up with a 2min arc gap thats how large a pattern we need to be able to resolve pattern VA is 6/12 - this is because quality of image is affected by aberrations so we are no longer affected by diffraction - we can only see what is allowed by retina and retina depends on light level