Lecture 27, and the last one

Lecture 27, and the last one

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iClicker

§ Are you here?

§ A: Yes.

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Wave-particle duality

§ Quantum mechanics: Light is both a wave and a particle• More of a wave at low frequencies (radio - optical)• More of a particle at high frequencies (X and gamma

rays)

§ Particle: propagates like a ray§ Wave: diffraction, interference - THE PHASE!

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Review: double slit

§ Young’s double slit interference experiment:

§ Monochromatic light§ Coherent light§ Huygens’ wavelets expand from

the slits§ Wavelets interfere§ Pattern on the distant screen:

Bright fringes (constructive interference)Dark fringes(destructive interference)

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Double Slit Intensity (1)

§ We can not only calculate the location of the fringes but also their intensity

§ We start our calculation of the intensity of light from a double slit by assuming that the light emitted at each slit is in phase

§ The electric field of the light waves can be described by

where Emax is the amplitude of the wave and ω is the angular frequency

§ When the light waves arrive at the screen from the two slits, they have traveled difference distances, and so can have difference phases that depend on the angle of observation

E = E

max

sin(!t� kx)

k =2⇡

�

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Double Slit Intensity (2)

§ Let’s express the electric field of the light arriving at a given point on the screen (fixed x) from S1 as

and the electric field of the light arriving at the same point from S2 as

where φ is the phase constant of E2 with respect to E1The magnitude of the sum of the two phasors is

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Electric Field Intensity

§ We know that the intensity of an electromagnetic wave is proportional to the square of the electric field so we can write

which gives us the relative intensity of the summed wave as a function of the phase difference between the two light waves

§ Now we need to relate the phase difference to the path length difference

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Double Slit Intensity on a Distant Screen (1)

§ Looking at our previous figurewe can see that the path lengthdifference Δx will cause aphase shift given by

§ Remember that§ So we can write the phase constant as

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§ Thus we can write an equation for the intensity of the light produced by the interference from two slits as

§ Because we are assuming that we are looking at the intensity pattern in a distant screen we can make the approximation

§ And we get an expression for the intensity

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§ The graph below represents the intensity a distance L away from the slits, where the slits are separated by d = 10-5 m and the wavelength of the incident light is λ = 550 nm

§ This picture is true only for y << L

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Double Slit Intensity on a Distant Screen

§ This picture is true only for y << L

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Double Slit Intensity on a Distant Screen

-30 -20 -10 10 20 30

-1.0

-0.5

0.5

1.0

y ~ L

Pattern for single slitdiffraction

Pattern for double slitinterference

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Double Slit Interference + Diffraction

convolution of diffraction and interference

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Double Slit Interference + Diffraction (2)

§ Including diffraction effects, the intensity of the interference pattern from double slits is given by

§ If the screen is placed a sufficiently large distance from the slits then we can write

recall: a is size of the slit, d is distances between slits

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Real Life Two Slit Diffraction Pattern

Diffraction minima

�y =�L

⇡d

�y =�L

⇡a

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Example: Two Slit Diffraction Pattern (1)

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Summary - Single and Double Slits

Double Slit (slit width a<<λ) Single Slit (slit width a~ λ)

Double Slit (slit width a~λ)

Reflection grating

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Note: in ray optics this would have produced a single image by tilted mirror. In wave optics we get diffraction pattern

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Diffraction Gratings (1)

§ We have discussed diffraction and interference for a single slit and for two slits

§ Now we will discuss the application of diffraction and interference to a system of many slits

§ Putting many slits together forms a device called a diffraction grating

§ A diffraction grating has a large number of slits, or rulings, placed very close together

2µm

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Diffraction Gratings (2)§ A diffraction grating can also be constructed using an

opaque material with grooves rather than actual slits§ A diffraction grating produces an intensity pattern that

consists of narrow bright fringes separated by wide dark areas

§ This characteristic pattern results from the use of many slits that produce destructive interference away from the maxima

2µm

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§ A portion of a diffraction grating is shown below§ In this drawing we see coherent light with wavelength

incident on a series of narrow slits each separated by a distance d

§ A diffraction pattern is produced on a screen a long distance L away

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Diffraction Gratings (5)§ We can calculate the path length differences for

the paths shown on the right§ Using an adjacent pair of rays, the path length

difference is

§ To produce bright lines or constructive interference this path length difference must be an integer multiple of the wavelength so

§ The values of m correspond to different bright lines• m = 0: central maximum at θ = 0

• m = 1: the first order maximum• m = 2: the second order maximum, etc.

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§ Typically diffraction gratings are designed to produce large angular separations between the maxima so we do not make the small angle approximation when discussing diffraction gratings

§ Because diffraction gratings produce widely spaced narrow maxima, they can be used to determine the wavelength of monochromatic light by rearranging

§ Monochromatic light incident on a diffraction grating will produce lines on a screen at widely separated angles

Turbulent refraction

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“Twinkle little star”

Previous exams

✓

sin ✓ = nwater/nglass = cos�

� = 29.0

p = I/c

F = pA

A = ⇡r2

a = F/m

m =4

3⇡r3⇢

a =3

4

I

r⇢c

a = 1.5⇥ 10�9

✓1

✓2

x1

x2

l1

l2

x = x1 + x2

x1 = l1 cot ✓

x2 = l2 tan ✓2

sin ✓2 = sin ✓1/n

The end...

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Energy!

l

U =1

2

X

i

qV

U1 =3

2kq2

l

U2 = k

✓1

2

q2

l/2+

1

2

q2

l/2+

1

2

q2

l

◆=

5

2kq2

l

U2 � U1 = kq2

l= Wt

�U1�2 =1

2kq2

l/2

�U1�3 =1

2kq2

l/2

�U = �U1�2 +�U1�3 = kq2

l

B = µ0nI

n =

B

µI

one loop : l = ⇡d

N = Lsn

Lw = lN = ⇡dLsB

µI

Ls - length of solenoidd - diameter of solenoidLw- length of wire

Ls

d

B = µ0nI

n =

B

µI

one loop : l = ⇡d

N = Lsn

Lw = lN = ⇡dLsB

µI

Lecture 27, and the last one

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Transcript of Lecture 27, and the last one