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Homework #5

Homework #5. Due 2/26/13. Problem 1. Eye resolution. Photoreceptor spacing This sets best possible resolution Angle = 2s/f and spatial frequency v = f/2s Pupil diffraction can degrade receptor resolution Frequency = 1/w = D /λ So D/λ ≥ f/2s. Problem 1. Angle.

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Homework #5

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  1. Homework #5 Due 2/26/13

  2. Problem 1. Eye resolution • Photoreceptor spacing • This sets best possible resolution • Angle = 2s/f and spatial frequency v = f/2s • Pupil diffraction can degrade receptor resolution • Frequency = 1/w = D /λ • So D/λ ≥ f/2s

  3. Problem 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 resolution • D/λ ≥ 4000 so • D ≥ 4000 * 500 x 10-9 m * 1e3 mm / m = 2 mm

  4. 1c 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

  5. 1d. What a bionic receptor sees E 0.5 m 16mm So x = 0.021m = 2.1 cm

  6. So if letters are big enough you could read 1-2 letters at a time 11cm 2.2 2.2 6.6cm

  7. Problem #2 : Reflection 2a. Reflection at water / cornea 2b. Reflection at aqueous humour / lens So no reflective losses since indices are so similar

  8. Problem #1 : Refraction 2c. Refraction at water / cornea interface 2d. Refraction at aqueous humour / lens 2e. So lens does the most bending!

  9. Compare terrestrial vs aqueous vision

  10. Sensitivity and resolutionD and d are not the sameD=aperture diameterd=receptor diameter • Sensitivity S = 0.62 D2 Δρ2 Pabs • Resolution, 1/Δρ = f/d focal length / receptor diam

  11. Problem 3a • Resolution = 1/Δρ= f/d • Sensitivity = 0.62 D2 Δρ2 Pabs • = 0.62 D2 d2 Pabs / f2 • Increasing lens focal length, f • Increases resolution f x2 resolution x2 • Decreases sensitivity f x2 sensitivity x1/4

  12. Problem 3b • Resolution = 1/Δρ= f/d • Sensitivity = 0.62 D2 Δρ2 Pabs • = 0.62 D2 d2 Pabs / f2 • Increasing photoreceptor diameter, d • Decreases resolution d x2 resolution x1/2 • Increases sensitivity d x2 sensitivity x4

  13. Problem 3c • Resolution = 1/Δρ= f/d • Sensitivity = 0.62 D2 Δρ2 Pabs • = 0.62 D2 d2 Pabs / f2 • Increasing eye aperture, D • No effect on resolution (except in diffraction limit) • Increases sensitivity D x2 sensitivity x4

  14. 3c caveat • Increasing D does not affect resolution • Decreasing D will get into diffraction limit • Diffraction angle w = λ/D • Diffraction limited resolution = 1/w=D / λ • Receptor defined resolution = f / d

  15. Receptor determined resolution f=20 mm d=2 um

  16. Diffraction will decrease resolution where diffraction is limiting f=20 mm d=2 um

  17. Problem 3d • To increase Pabs • A = 1 - T = 1 - exp(-εCl) • As εCl gets large • exp(-εCl) gets small • A gets large

  18. Problem 3d • Increase εCl • ε = extinction coefficient • Property of molecule - could put in a more strongly absorbing pigment but probably not easy to vary • C = concentration • Pack more pigment into receptor • l = receptor length • Make receptor longer

  19. Problem 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

  20. Problem 3e • How can diurnal organisms increase resolution? • Resolution = 1/Δρ = f / d • Increase lens focal length • Decrease receptor size

  21. Problem 3f • How can nocturnal organisms increase sensitivity? • S = 0.62 D2 Δρ2Pabs Δρ = d/f • Increase pupil / lens diameter, D • Increase receptor diameter, d • Decrease lens focal length, f • Make receptors longer

  22. 4. Number of photons • A) Photons reaching receptor

  23. If photoreceptor only absorbs 30% of photons • Number of photons detected = • 30% of 300 photons / s = 90 photons / s • Minimum detectable contrast is 10.5%

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