Diffraction: single slit

How can we explain the pattern from light going through a single slit?

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Diffraction: single slit

If we break up the single slit into a top half and a bottom half, then we can consider the interference between the two halves.

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Diffraction: single slit

The path difference between the top half and the bottom half must be /2 to get a minimum.

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Diffraction: single slit

This is just like the double slit case, except the distance between the “slits” is w/2, and this is the case for minimum: (w/2) sin() = /2

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Diffraction: single slit

In fact, we can break the beam up into 2n pieces since pieces will cancel in pairs. This leads to: (w/2n) sin(n) = /2 ,

or w sin(n) = n for MINIMUM.

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Diffraction: single slit

REVIEW:

• For double (and multiple) slits:

n = d sin(n) for MAXIMUM

(for ALL n)

• For single slit:

n = w sin(n) for MINIMUM

(for all n EXCEPT 0)

Diffraction: single slit

NOTES:• For double slit, bright spots are equally

separated.• For single slit, central bright spot is larger

because n=0 is NOT a dark spot.• To have an appreciable , d and w must be

about the same size as & a little larger than • Recall that for small angles, sin) = tan() = x/L

Diffraction: circular opening

If instead of a single SLIT, we have a CIRCULAR opening, the change in geometry makes:

• the single slit pattern into a series of rings; and

• the formula to be: 1.22 n = D sin(n) • CH 5-4 shows the rings in several diagrams

and the use of this equation.

Diffraction: circular opening

Since the light seems to act like a wave and spreads out behind a circular opening, and since the eye (and a camera and a telescope and a microscope, etc.) has a circular opening, the light from two closely spaced objects will tend to overlap. This will hamper our ability to resolve the light (that is, it will hamper our ability to see clearly).

Diffraction: circular opening

How close can two points of light be to still be resolved as two distinct light points instead of one? One standard, called the Rayleigh Criterion, is that the lights can just be resolved when the angle of separation is the same as the angle of the first dark ring of the diffraction pattern of one of the points: limit = 1 from 1.22 * = D sin(1) .

Rayleigh Criterion: a picture

The lens will focus the light to a fuzzy DOT

rather than a true point.

lensD

Rayleigh Criterion: a picture

The Rayleigh minimum angle,

limit = sin-1(1.22 /D) = x’ / s’.

lensD

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x’

Rayleigh Criterion: a picture

If a second point of light makes an angle of

limit with the first point, then it can just be resolved. lensD

s’

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Rayleigh Criterion: a picture

In this case:

limit = sin-1(1.22 /D) = x’/s’ = x/s .

lensD

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Rayleigh Criterion: an example

• Consider the (ideal) resolving ability of the eye

• Estimate D, the diameter of the pupil• Use = 550 nm (middle of visible spectrum)

• Now calculate the minimum angle the eye can resolve.

• Now calculate how far apart two points of light can be if they are 5 meters away.

Rayleigh Criterion: an example

• with D = 5 mm and = 550 nm,

limit = sin-1 (1.22 x 5.5 x 10-7 m/.005 m)

= 7.7 x 10-3 degrees

= .46 arc minutes

so x/L = tan(limit), and

x = 5m * tan(7.7 x 10-3 degrees) = .67 mm

Rayleigh Criterion: an example

• Estimate how far it is from the lens of the eye to the retinal cells on the back of the eye.

• With your same D and and so same limit),

now calculate how far the centers of the two dots of light on the retina are.

• How does this distance compare to the distance between retinal cells (approx. diameter of the cells)?

Rayleigh Criterion: an example

• L = 2 cm (estimation of distance from lens to retinal cells)

• from previous part, limit = 7.7 x 10-3 degrees

• so x = 2 cm * tan(7.7 x 10-3 degrees)

= 2.7 m .

Limits on Resolution:

• Imperfections in the eye (correctable with glasses)

• Rayleigh Criterion due to wavelength of visible light

• Graininess of retinal cells

Limits on Resolution: further examples

• hawk eyes and owl eyes

• cameras:– lenses (focal lengths, diameters)– films (speed and graininess)– shutter speeds and f-stops

• Amt of light D2 t• f-stop = f/D

– f-stops & resolution: resolution depends on D

Limits on Resolution: further examples

1.22 n = D sin(n) where 1 = limit

• microscopes: smallest size = = .5 m– can easily see .5 mm, so M-max = 1000– can reduce by immersing in oil, use blue light

Limits on Resolution: further examples: the telescope

Light from far away is almost parallel.

objectivelens

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Limits on Resolution: further examples

The telescope collects and concentrates light

objectivelens

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Limits on Resolution: further examples

Light coming in at an angle, in is magnified.

objectivelens

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Limits on Resolution: further examples

in = x/fo, out = x/fe; M = out/in = fo/fe

objectivelens

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fobjfeye

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Limits on Resolution: further examples

• telescopes– magnification: M = out/in = fo /fe

– light gathering: Amt D2

– resolution: 1.22 = D sin(limit) so

in = limit and out = 5 arc minutes

so limit 1/D implies Museful = 60 * D

where D is in inches– surface must be smooth on order of

Limits on Resolution: Telescope

• Mmax useful = out/in = eye/limit

= 5 arc min / (1.22 * / D) radians

= (5/60)*(/180) / (1.22 * 5.5 x 10-7 m / D)

= (2167 / m) * D * (1 m / 100 cm) * (2.54 cm / 1 in)

= (55 / in) * D

Limits on Resolution: TelescopeExample

• What diameter telescope would you need to read letters the size of license plate numbers from a spy satellite?

Limits on Resolution: TelescopeExample

• need to resolve an “x” size of about 1 cm

• “s” is on order of 100 miles or 150 km

• limit then must be (in radians)

= 1 cm / 150 km = 7 x 10-8

• limit = 1.22 x 5.5 x 10-7 m / D = 7 x 10-8

so D = 10 m (Hubble has a 2.4 m diameter)

Limits on Resolution: further examples

• other types of light– x-ray diffraction (use atoms as slits)– IR– radio & microwave

• surface must be smooth on order of

Polarization

• Experiment with polarizers

• Particle Prediction?

• Wave Prediction?– Electric Field is a vector: 3 directions

• Parallel to ray (longitudinal)• Maxwell’s Equations forbid longitudinal

• Two Perpendicular (transverse)

Polarization: Wave Theory

• #1 Polarization by absorption

(Light is coming out toward you)

unpolarized polarizeronly letsvertical componentthrough

polarizedlight polarizer

only letshorizontal componentthrough

no lightgetsthrough

Polarization: Wave Theory

• Three polarizers in series:

Sailboat analogy:

North wind

sail

force onsail boat goes along

direction of keel

Polarization: Wave Theory

• #2 Polarization by reflection– Brewster Angle: when refracted + reflected = 90o

– Sunglasses and reflected glare

surface

incident rayvertical

horizontal

refracted ray

reflected rayno problem with horizontal

almost no vertical since vertical is essentially longitudinal now

vertical can be transmitted

Polarization: Wave Theory

• #3 Polarization by double refraction– different n’s in different directions due to

different bonding

• #4 Polarization by scattering