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Optics 2----by Dr.H.Huang, Department of Applied Physics 1
The Hong Kong Polytechnic University Diffraction
Introduction:
Diffraction is often distinguished from interference on that: in diffraction phenomena, the interfering beams originate from a continuous distribution of sources; in interference phenomena the interfering beams originate from a discrete number of sources.
If both the source of light and observation screen are effectively far enough from the diffraction aperture so that the wavefronts arriving at the aperture and observation screen may be considered plane, it is called Fraunhofer, or far-field, diffraction. When the curvature of the wavefront must be taken into account, it is called Fresnel, or near-field, diffraction.
Optics 2----by Dr.H.Huang, Department of Applied Physics 2
The Hong Kong Polytechnic University Diffraction
Fraunhofer Diffraction at Single Apertures:
Each interval contributes spherical wavelets at P of the form,
tkrip e
r
dEdE
0
trkip e
r
dEdE
00
tkskriLtrkiLp e
r
dsEe
r
dsEdE
sin
0
00
tkrib
b
iksLp edse
r
EE
0
2
2
sin
0
2
2
0
2 sin
IEI p
sin2
1kb
Optics 2----by Dr.H.Huang, Department of Applied Physics 3
The Hong Kong Polytechnic University Diffraction
Fringe Pattern:
Dark fringe:
The second, third and fourth maxima of the diffraction pattern occur at =1.43, 2.46 and 3.47, respectively.
mkb sin2
1
b
fmy
The central maximum represents essentially the image of the slit on a distant screen.
The angular width of the central maximumIs
The linear width of the central maximum is
b
2
b
LLW
2
Optics 2----by Dr.H.Huang, Department of Applied Physics 4
The Hong Kong Polytechnic University Diffraction
Rectangular Slits:
2
2
2
2
0
sinsin
II
b
fmy
a
fnx
b
a
a
Optics 2----by Dr.H.Huang, Department of Applied Physics 5
The Hong Kong Polytechnic University Diffraction
Circular Slits:
The far-field angular radius of Airy disc is,
2
10
)(2
J
II
sin2
1kD
22.1sin D
3.832
Airy Disc
D
22.1
Optics 2----by Dr.H.Huang, Department of Applied Physics 6
The Hong Kong Polytechnic University Diffraction
Rayleigh’s Criterion:
For a microscope,
The ratio D/f is the numerical aperture.
D
22.1)( min
Dffx
22.1minmin
Optics 2----by Dr.H.Huang, Department of Applied Physics 7
The Hong Kong Polytechnic University Diffraction
Example:Two stars have an angular separation of 44.7310-7 radian. Find the minimum diameter of the telescope objective which can just resolve the stars in light of 550 nm wavelength.
Example:Calculate the minimum angular subtense of two points which can be just resolved by an eye with a 6 mm diameter pupil in light of 555 nm wavelength.
cm151073.4422.1 7min d
d
radian10129.122.1 4min
d
Optics 2----by Dr.H.Huang, Department of Applied Physics 8
The Hong Kong Polytechnic University Diffraction
Diffraction by Small Particles:
Babinet’s Principle
1 and 2 are complementary apertures.
Suppose that monochromatic plane wavefronts are incident normally on 1 and the diffracted light is imaged on a screen.
In a direction to the normal let the magnitude of the electric vector be E1. Replace 1 with 2 and let the magnitude of the
electric vector be E2. Apparently,
E1+E2=0 and E1=E2
Therefore,
It means that the diffraction patterns for 1 and 2 are identical.
21 II
Optics 2----by Dr.H.Huang, Department of Applied Physics 9
The Hong Kong Polytechnic University Diffraction
Fraunhofer Diffraction at Two Slits:
ba
ba
iskba
ba
iskLR dsedse
r
EE
21
21
sin21
21
sin
0
2
2
0 cossin
4
II
sin2
1kb sin
2
1ka
sin
cos2
sincoscos 222 aka
a=2b
Optics 2----by Dr.H.Huang, Department of Applied Physics 10
The Hong Kong Polytechnic University Diffraction
Fraunhofer Diffraction at Two Slits:
a=6b
2
2
0 cossin
4
II
Optics 2----by Dr.H.Huang, Department of Applied Physics 11
The Hong Kong Polytechnic University Diffraction
Diffraction at Many Slits:
2
1
212
212
sin212
212
sin
0
N
j
baj
baj
iskbaj
baj
iskLR dsedse
r
EE
22
0 sin
sinsin
N
II
N=8 and a=3b
ma sinprincipal maxima at
Optics 2----by Dr.H.Huang, Department of Applied Physics 12
The Hong Kong Polytechnic University Diffraction
Diffraction Grating:
ma mi sinsin
Optics 2----by Dr.H.Huang, Department of Applied Physics 13
The Hong Kong Polytechnic University Diffraction
Free Spectral Range:—the non-overlapping wavelength range in a particular order.
The non-overlapping spectral region is smaller for higher orders.
The wavelength are better separated as their order increases. This property is described by angular dispersion, or dispersive power of a grating,
Linear dispersion is,
Resolving power of a grating is defined as,
Using the Rayleigh’s criterion, suppose the number of grating grooves is N, we have
ma sin
d
dD m
ma
mD
cos
fDd
df
d
dy m
min
R
mNR
Optics 2----by Dr.H.Huang, Department of Applied Physics 14
The Hong Kong Polytechnic University Diffraction
Example:A grating has 4000 grooves or lines per centimeter. Calculate the dispersive power in the second order spectrum in the visible range.
We take the mean wavelength to be 550 nm.
Example:Find the number of lines (grooves) required on a grating to just resolve the two sodium lines, 1=589.592 nm and 2=588.995 nm, in the second order spectrum of a grating.
rad/m109.8cos
10.26sin2;m105.24000
1
5
2
26
a
mD
mama m
494
2
min
21min21
NmNR
Optics 2----by Dr.H.Huang, Department of Applied Physics 15
The Hong Kong Polytechnic University Diffraction
Fresnel Diffraction:
ikrp e
r
dEdE
0
daEdE L0
rikSL e
r
EE
dae
rr
EdE rrikS
p
darr
eF
ikEE
rrikS
p 2
201
rr
02 rr
20
NrrN
Optics 2----by Dr.H.Huang, Department of Applied Physics 16
The Hong Kong Polytechnic University Diffraction
...
...
4321
34
2321
aaaa
eaeaeaaA iii
11 aA
212 aaA
3213 aaaA
Two conclusions
(1) If N is small, there is large changes in the resultant phasor AN as the contribution
from each new zone is added. The resultant amplitude seems to oscillate between magnitudes that are larger and smaller than the limiting value of a1/2. As the
aperture gradually increases, one can see oscillations between bright and dark in a fixed position of the screen.
(2) If N is large, as in the case of unlimited aperture, the resultant amplitude is half that of the first contribution zone, a1/2.
Optics 2----by Dr.H.Huang, Department of Applied Physics 17
The Hong Kong Polytechnic University Diffraction
Fresnel zone plate:very other Fresnel zone is blocked
The zone plate radii are approximately given by, 0NrRN
Optics 2----by Dr.H.Huang, Department of Applied Physics 18
The Hong Kong Polytechnic University Diffraction
Diffraction by Straight Edges :Use cylindrical waves.
Optics 2----by Dr.H.Huang, Department of Applied Physics 19
The Hong Kong Polytechnic University Diffraction
Example:Plane wave of =550 nm are incident normally on a circular aperture of radius mm. Does a bright or a dark spot appear at the point P on the axis 4 m from the hole? If the intensity of the incident light is I0, calculate the intensity at P.
Example:A 4 mm diameter circular hole in an opaque screen is illuminated by plane waves of wavelength 500 nm. If the angle of incidence is zero, find the positions of the first two intensity maxima and the first intensity minimum along the central axis.
The first two maxima will occur when N=1 and 3, respectively. The first minimum occurs when N=2.
11
0
2
1
1
0154321
0
2
0
442
spotbrightnumberodd5
IIE
E
I
IEEEEEEE
r
RNNrR N
N
m4:2
m67.2:3m8:1
2
22
N
RrN
N
RrN
N
RrN
Optics 2----by Dr.H.Huang, Department of Applied Physics 20
The Hong Kong Polytechnic University Diffraction
Example:Plane waves of 550 nm wavelength are incident normally on a narrow slit of width 0.25 mm. Calculate the distance between the first minima on either side of the central maximum when the Fraunhofer diffraction pattern is imaged by a lens of focal length 60 cm.
Example:Plane waves (=550 nm) fall normally on a slit 0.25 mm wide. The separation of the fourth order minima of the Fraunhofer diffraction pattern in the focal plane of the lens is 1.25 mm. Calculate the focal length of the lens.
Example:Light from a distant point source enters a converging lens of focal length 22.5 cm. How large must the lens be if the Airy disc is to be 10-6 m in diameter? =450 nm
mm64.22
b
ffW
cm10.78
88sin
2
1
WbW
fb
kb
cm7.2410
44.210
22.122
66
fD
Dff
Optics 2----by Dr.H.Huang, Department of Applied Physics 21
The Hong Kong Polytechnic University Diffraction
Example:A telescope objective is 12 cm in diameter and has a focal length of 150 cm. Light of mean wavelength 550 nm from a star is imaged by the objective. Calculate the size of the Airy disc.
Example:Assuming Rayleigh’s criterion can be applied to the eye, how far apart must two small lights be in order to be just resolved at a distance of 1000 m? Take the pupil diameter as 2.5 mm, the wavelength to be 555 nm, and the eye’s refractive index 1.333. Assume a single surface model eye with the pupil at the surface.
mm017.022.1
22 D
ff
minmin
min 2sin
2sin
22.122.1
nnnDD i
im
cm1.2722.1
minmin D
LLnLx i