Chapter 24:Wave Optics Conditions for Interference Homework assignment : 10,22,23,33,42,55 ...
-
Upload
melvin-caldwell -
Category
Documents
-
view
217 -
download
0
Transcript of Chapter 24:Wave Optics Conditions for Interference Homework assignment : 10,22,23,33,42,55 ...
Chapter 24:Wave OpticsConditions for Interference
Homework assignment : 10,22,23,33,42,55
Conditions for interference
Adding two waves: wave 1 and wave 2
Conditions for interference (cont’d)
• The sources must be coherent, which means the waves they emit must maintain a constant phase with respect to each other.• The waves must have identical wavelengths.
• Two sources producing two traveling waves are needed to create interference.
An old style method to produce two coherent sources
Conditions for Interference
Two slits and interference
Conditions for Interference
Two monochromatic sources of the same frequency and with anyConstant phase relation are said to be coherent.
Young’s double-slit experiment
Young’s Double-Slit Experiment
• Red dots represent spots where the two waves are in phase and constructive interference occurs. This produces bright spots.
• Blue circles represent spots where the two waves are out of phase and destructive interference occurs. This produces dark spots.
Young’s double-slit experiment (cont’d)
Young’s Double-Slit Experiment
In phase
Bright image
In phase
Bright image
Out of phase
Dark image
Young’s double-slit experiment (cont’d)
Young’s Double-Slit Experiment
sin12 drr ,...)2,1,0( sin mmd bright
Path difference :
Constructive interference:
m : order number m=0 zeroth-order, m=+-1 first-order…
Destructive interference : ,...)2,1,0( )2
1(sin mmd bright
Young’s double-slit experiment (cont’d)
Young’s Double-Slit Experiment
1 if sintan LLy
,...)2,1,0( mmd
Lybright
Constructive interference:
Destructive interference: ,...)2,1,0( )2
1( mm
d
Lydark
Example 24.1: Measuring the wavelength of a light source• A screen is separated from a double-slit source by 1.20 m. The distance between the two slits is 0.0300 mm. The second-order bright fringe (m=2) is measured to be 4.50 cm from the centerline. Determine: (a) the wavelength of the light
(b) the distance between the adjacent bright fringes.
m 1063.5 72 mL
dy
cm 25.2)1(1 d
Lm
d
Lm
d
Lyyy mm
Young’s Double-Slit Experiment
Lloyd’s mirror• In the arrangement of a mirror and the screen on the right, at a point on the screen the light waves have two possible paths.
Change of Phase Due to Reflection
• At point P’, the path difference of two paths is zero: Therefore we expect a bright fringe. But we see a dark fringe there.• Similarly on the screen the pattern of bright and dark fringes is reversed.
An EM wave undergoes a phase change of 180o or upon reflectionfrom a medium that has an index of refraction higher than the one inwhich the wave is traveling.
Phase change upon reflection
Change of Phase Due to Reflection
An EM wave undergoes a phase change of 180o upon reflection froma medium that has an index of refraction higher than the one in whichthe wave is traveling.
Colorful patterns
Interference in Thin Film
Interference in thin film
Interference in Thin Film
• An EM wave traveling from a medium of index of refraction n1
toward a medium of index of refraction n2 undergoes a phase change on reflection when n2>n1. There is no phase change in the reflected light wave if n1>n2.• The wavelength of light n in a medium with index of refraction n is n=/n where is the wavelength of light in vacuum.Ray 1: A phase change of at surface A with respect to the incident wave (n>nair=1)
Ray 2: No phase change at surface B with respect to the incident wave (n>nair=1)
pha
se d
iff. b
y
due
to r
efle
ctio
n
Phase difference Ray 1 : Ray 2 : 2 x 2t/n
)/2(212 nt
.)0,1,2,3,..(m )2( m
1,2,3,...),0(m )2)(2/1( m
constructive
destructive
Interference in thin film (cont’d)
Interference in Thin Film
Interference between ray 1 and ray 2
,...)2,1,0( )2
1(2 mmt n
,...)2,1,0( 2 mmt n
constructive
destructive
)2
1(2 mnt
mnt 2
thicker dueto gravity
Newton’s Rings
Interference in Thin Film
• Another example of interference: A planoconvex lens on top of a flat glass surface
Blue ray : phase change at the interface between air and a flat glass.Red ray : no phase change due to reflection.
pha
se d
iff. b
y
due
to r
efle
ctio
n
Interference between red and blue rayconstructive
destructive
RdRdRRdRr
RddRRr
2)/21(1)/1(1
)(
2
22
0,1,2,...)(m )2/)(2/1( mdm
0,1,2,...)(m )2/( mdm
Non-reflective coating for optical lens
Interference in Thin Film
,...)2,1,0( 2 mmt n
)2
1(2 mnt
MgF2 n = 1.38
Glass, n = 1.50
Air, n = 1.00
1 2
t
Thin layer of MgF2 with index intermediate between air and glass;there are two phase shifts of in this case, as shown.Phase shifts of reflected rays:
Ray 1: Ray 2: + (2/n)(2t)
For no reflection: (n) (2t) =
ntt
n 42
/
2
Numbers: for no reflection at = 550 nm, t = 100 nm.
(destructive interference)
Interference in a wedge-shaped film
Interference in Thin Film
A pair of glass slides 10.0 cm longand with n=1.52 are separated onone end by a hair, forming a triangularwedge of air as illustrated. Whencoherent light from a helium-neonlaser with wavelength 633 nm isincident on the film from above, 15.0dark fringes per cm are observed.How thick is the hair?For destructive interference with nair=1: 2/mt If d is the distance from one dark bandto the next, the x-coord. of the m-th fringe is: mdx s)(cm/fringe 1067.6)fringes/cm 0.15( 2-1 d
Using similar triangles : LDdmdmxt /)2/()/()2/(/
m 1075.42
5d
LD
What is diffraction?
Diffraction
Diffraction is bending of waves around obstacles. Diffractionalso occurs when waves from a large number of sources interfere.
Diffraction Single slit
Bending on the corner
Diffraction at slits Single slit
•
•
Diffraction and Huygens principleEvery point on a wave front acts as a source of tiny wavelets that move forward.
We will see maxima and minima on the wall.
Light waves originating at different points within opening travel different distances to wall, and can interfere!
Diffraction at slits
Diffraction at slits Single slit
1
1
11
sin4
a
= path difference
= path difference
When a/2 sin = /2 every ray originating in top half of slit interferes destructively with the corresponding ray originating in bottom half.
When a/4 sin = /2 every ray originating in top quarter of slit interferes destructively with the corresponding ray originating in the second quarter.
a2
a
sin2
a
4
a
a
Diffraction at slits Single slit
Condition for quarters of slit to destructively interfere
Condition for halves of slit to destructively interfere a
sin
a
2sin
Condition for sixths of slit to destructively interferea
3sin
This formula locates
minima (for dark fringes)!!
All together… ,...)3,2,1( sin ma
mdark
Diffraction Grating Diffraction grating• Combination of diffraction and interference• Consists of a large number of equally spaced parallel slits.• Can be made by scratching parallel lines on a glass plate with a precision machine technique.
Each slit causes diffraction. Each slit acts as a source of waves in phase. Path diff. between two adjacent slits at P: sind
d=spacing of slits If slits are made to satisfy :
waves from slits are in phase.,...)2,1,0( sin mmd bright
bright are the positions of bright maxima.
Diffraction Grating Diffraction grating (cont’d)
Pattern of light intensity Diffraction grating spectrometer
By measuring the positions of maxima, using the telescope in a diffractiongrating spectrometer, the wavelength of the incident waves can bedetermined.
Polarization
EM wave
BE
tkxEjtxB
tkxEitxE
)cos(ˆ),(
)cos(ˆ),(
max
max
Polarization (defined by the direction of )E
Linear polarization Circular polarization
vector inx-dir.vector iny-dir.
If the E-field is in the same directionall the time, the EM wave is linearly polerized.
Polarization (cont’d)
Polarization (defined by the direction of )E
Circular polarization
Polarization (cont’d) Polarization by reflection
p p
b
an
bn
plane of incidenceWhen the angle of incident coincides withthe polarizing angle or Brewster’s angle,the reflected light is 100% polarized.
p
pbpbbbpa nnnn cos)90sin(sinsin
a
bp n
ntan Brewsters’ law of the polarizing angle
Polarization (cont’d)
Polarization by absorption and polarizing filters
Polaroid: polarizes light through selective absorption by oriented molecules
These molecules have a preferred axis (transmission axis) along which electronscan move easily. All lights with E parallel(perpendicular) to the axis is transmitted(absorbed).
• Unpolarized light (like the light from the sun) passes through a polarizing sunglass (a linear polarizer). The intensity of the light when it emerges is:
20
2 cosIIEI
Malus’s Law
Malus’s law
• When linearly polarized light passes through a polarizer whose axis
of transmission make an angle with the direction of the E-field vector (or polarization), the intensity of the light follows the law:
02
2
1IIEI
Malus’s Law
Malus’s law (cont’d)
2max cosII
Malus’ law for polarizedlight passing through ananalyzer/polarizer:
This is true only whenthe incident light islinearly polarized.
Unpolarized light on linear polarizer
• Most light comes from electrons accelerating in random directions and is unpolarized.
• Averaging over all directions, intensity of transmitted light reduces due to reduction in E
Malus’s Law
Example:Malus’s law
unpolarized light
E1
I = I0
TATA
TA
E0
I3
B1
unpolarized light
E1
I = I0
TATA
TA
E0
I3
B1
1) Intensity of unpolarized light incident on linear polarizer is reduced by half . I1= I0 / 2
2) Now ignore the 2nd polarizer for now.
I = I0I1
I3
3) Light transmitted through first polarizer is vertically polarized. Angle between it and 3rd polarizer is =90º . I3= I1 cos2(90º) = 0
Malus’s Law
Example: Malus’s law
2) Light transmitted through first polarizer is vertically polarized. Angle between it and second polarizer is =45º. I2= I1 cos2 (45º)=0.5I1=0.25 I0
3) Light transmitted through second polarizer is polarized 45º from vertical. Angle between it and third polarizer is =45º. I3= I2 cos2(45º)
I2= I1cos2(45)
=0.125 I0
I1= 0.5 I0
unpolarized light
E1
I = I0
TATA
TA
E0
I3
B1
unpolarized light
E1
I = I0
TATA
TA
E0
I3
B1
1) Now put 2nd polarizer in.
Malus’s Law
I1 I2
Example: Malus’s law
Malus’s Law
Controlled Polarization
• Electrical voltage on a liquid crystal diode– Turns on and off
polarizing filter effect– Used in LCD display
LCD screen