Optics III: Diffraction

Light passing a horizontal, single slit produces…

Diffraction pattern perpendicular to the slit

Horizontal bands spreading vertically

Maximum intensity in the single slit experiment

At the center, the following dark bands are the darkest

Following light/dark bands are less bright/dim

Huygens-Fresnel Principle

Every point on a wavefront is considered a source of spherical diffraction

All points cause interference OR coherence on one another

What points make angle α

Points consists of…

• Center of the slit

• Center of the viewing screen

• Point being measured for angle α (Changes depending on where on the pattern you are measuring)

Calculating angular distance of a slit

• Angular distance

  • ∝ or θ

  • Measured from peak of bright center → Center of first dark band

• Equation variables

  • d = width of the slit

Finding path difference

△P = P1-P2

△P = d * sin * θ

• You’re looking at a point X

  • You wanna find the difference between the hypotenuse of the path travelled to point X from…

    • Top and bottom of the slit (Picture)

What’s the importance of path difference?

• When △P is a multiple of a full wavelength → Destructive interference

  • → d * sin(θ) = mλ

Diffraction pattern of square aperature

Diffraction pattern of circular aperture

• Bright, central disc surrounded by dark/bright rings

• Airy disk!

Obstacle vs. Slit diffraction pattern

Called reciprocity

• If the slit was instead an obstacle of the same size

  • Both spread light perpendicular to the slit/obstacle

    • Horizontal slit → Spreads vertically

    • Horizontal obstacle → Spreads vertically

Diffraction of astronomical telescope

• Shape

• How is it produced

• Stars produce a spiky diffraction pattern

• Diffraction produced by the secondary mirror

Calculating angular half-diameter (ɑ) of circular aperture

• Consists of center → Center of first dark ring

What exactly is an Airy’s disc

• Circle with diameter across the first dark ring

  • Includes entire bright circle and half of first dark ring

Rayleigh criterion

• Two point sources can be resolved as seperate entities if…

  • Their Airy discs are half-overlapped _{(Or less)}

Resolving power of any instrument with circular aperture…

• Equal to angle ɑ

  • Inversely proportional to instrument aperture diameter (d)

Conversion factor degree ↔ radians

57 deg/rad

Conversion factor arc mins ↔ deg

60 arc mins / deg

Conversion arc sec ↔ min

60 sec/min

Explain our eyes limitation to 20/20 vision

• Eye resolving power: 1.9×10^-4 rad

• 20/20 letter: 5 arc min total / 1 arc min for each detail = 2.9×10^-4 rad

Wait, we have plenty of resolving power?? Why not more than 20/20

• Foveal cone size: 3×10^-3 mm BUT distance to lens = 17mm

  • → 3×10^-3 mm / 17mm = 1.7×10^-4

    • About equal to resolving power of the eye

Size of pinhole camera aperture affects on…

• Quality

• Brightness

• Diffraction effects

Smaller the aperture…

• Resolution increases

  • Remember resolving power inversely proportional to aperture diameter

• Brightness decreases

• Diffraction increases

Diffractive lenses

• Front surface → Refractive surface _{(For distance vision)}

• Back surface

  • Grooved

  • Produces diffraction when tear film fills the grooves

    • → Helps with near vision

    • → Makes pupil fluctuation obsolete

    • → Reduction of contrast