Homework #5Homework #5

Due 2/26/13

Problem 1. Eye resolutionProblem 1. Eye resolution

• Photoreceptor spacing This sets best possible resolutionAngle = 2s/f and spatial frequency v =

f/2s

• Pupil diffraction can degrade receptor resolutionFrequency = 1/w = D /λ

So D/λ ≥ f/2s

Problem 1. AngleProblem 1. Angle

• Photoreceptor angle, 2s /f = 2*2um/16,000um = 0.00025 rad =

0.014°

Spatial frequency = 4000 cycles / rad

• For pupil not to limit this resolutionD/λ ≥ 4000 soD ≥ 4000 * 500 x 10-9 m * 1e3 mm / m = 2 mm

1c Bionic eye1c Bionic eye

• Each pin stimulates an area equal to spacing btn pins

= 7 mm / 10 pins = 0.7 mm• Resolution = f / 2s= 16 mm / 2*0.7mm= 11.4 cycles / rad

which is nearly 350x worse than a normal eye

1d. What a bionic receptor sees1d. What a bionic receptor sees

0.5 m

16mm

So x = 0.021m = 2.1 cm

So if letters are big enough you So if letters are big enough you could read 1-2 letters at a timecould read 1-2 letters at a time

2.2

2.26.6cm

11cm

Problem #2 : ReflectionProblem #2 : Reflection

2a. Reflection at water / cornea

2b. Reflection at aqueous humour / lens

So no reflective losses since indices are so similar

Problem #1 : RefractionProblem #1 : Refraction

2c. Refraction at water / cornea interface

2d. Refraction at aqueous humour / lens

2e. So lens does the most bending!

Compare Compare terrestrial terrestrial vs aqueous vs aqueous

visionvision

Reflection at cornea

0.025 0.00029

Reflection at lens

0.00065 0.0055

Refraction at cornea (24° in)

17.2° bends 6.8°

23.1° bends 0.9°

Refraction at lens (24° in)

22.7°bends 1.3°

20.5°Bends 3.5°

Sensitivity and Sensitivity and resolutionresolution

D and d are not D and d are not the samethe same

D=aperture diameterD=aperture diameterd=receptor diameterd=receptor diameter

• Sensitivity S = 0.62 D2 Δρ2 Pabs

• Resolution, 1/Δρ = f/d focal length / receptor diam

Problem 3aProblem 3a

• Resolution = 1/Δρ= f/d

• Sensitivity = 0.62 D2 Δρ2 Pabs

= 0.62 D2 d2 Pabs / f2

• Increasing lens focal length, fIncreases resolution f x2

resolution x2 Decreases sensitivity f x2 sensitivity

x1/4

Problem 3bProblem 3b

• Resolution = 1/Δρ= f/d

• Sensitivity = 0.62 D2 Δρ2 Pabs

= 0.62 D2 d2 Pabs / f2

• Increasing photoreceptor diameter, dDecreases resolution d x2 resolution

x1/2 Increases sensitivity d x2 sensitivity x4

Problem 3cProblem 3c

• Resolution = 1/Δρ= f/d

• Sensitivity = 0.62 D2 Δρ2 Pabs

= 0.62 D2 d2 Pabs / f2

• Increasing eye aperture, DNo effect on resolution (except in

diffraction limit)

Increases sensitivity D x2 sensitivity x4

3c caveat3c caveat

• Increasing D does not affect resolution

• Decreasing D will get into diffraction limitDiffraction angle w = λ/DDiffraction limited resolution =

1/w=D / λ

Receptor defined resolution = f / d

Receptor determined resolutionReceptor determined resolution

f=20 mmd=2 um

Diffraction will decrease resolution Diffraction will decrease resolution where diffraction is limitingwhere diffraction is limiting

f=20 mmd=2 um

Problem 3dProblem 3d

• To increase Pabs

A = 1 - T = 1 - exp(-εCl)As εCl gets largeexp(-εCl) gets smallA gets large

Problem 3dProblem 3d

• Increase εCl• ε = extinction coefficient

Property of molecule - could put in a more strongly absorbing pigment but probably not easy to vary

• C = concentrationPack more pigment into receptor

• l = receptor lengthMake receptor longer

Problem 3d caveatProblem 3d caveat

• Increasing aperture D will get more photons to receptor

• This is separate factor from Pabs

Sensitivity = 0.62 D2 Δρ2 Pabs

• Both work together to increase sensitivity

Problem 3eProblem 3e

• How can diurnal organisms increase resolution?Resolution = 1/Δρ = f / d

Increase lens focal lengthDecrease receptor size

Problem 3fProblem 3f

• How can nocturnal organisms increase sensitivity?S = 0.62 D2 Δρ2Pabs Δρ =

d/f

Increase pupil / lens diameter, D Increase receptor diameter, dDecrease lens focal length, fMake receptors longer

4. Number of photons4. Number of photons

• A) Photons reaching receptor

If photoreceptor only absorbs If photoreceptor only absorbs 30% of photons30% of photons

• Number of photons detected =30% of 300 photons / s = 90 photons /

s

• Minimum detectable contrast is 10.5%