Optics I

1
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.
2
Diffraction
Fraunhofer Diffraction at Single Apertures Each
interval contributes spherical wavelets at P of
the form,
3
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.
The central maximum represents essentially the
image of the slit on a distant screen. The
angular width of the central maximum Is The
linear width of the central maximum is
4
Diffraction
Rectangular Slits
a
a
b
5
Diffraction
Circular Slits The far-field angular radius
of Airy disc is,
Airy Disc
3.832
6
Diffraction
Rayleighs Criterion For a
microscope, The ratio D/f is the numerical
aperture.
7
Diffraction
Example Two stars have an angular separation of
44.73?10-7 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.
8
Diffraction
Diffraction by Small Particles Babinets
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,
E1E20 and E1?E2
Therefore, It means that the diffraction
patterns for ?1 and ?2 are identical.
9
Diffraction
Fraunhofer Diffraction at Two Slits
a2b
10
Diffraction
Fraunhofer Diffraction at Two Slits a6b
11
Diffraction
Diffraction at Many Slits
principal maxima at
N8 and a3b
12
Diffraction
Diffraction Grating
13
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 Rayleighs criterion,
suppose the number of grating grooves is N, we
have
14
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, ?1589.592 nm and
?2588.995 nm, in the second order spectrum of a
grating.
15
Diffraction
Fresnel Diffraction
16
Diffraction
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.
17
Diffraction
Fresnel zone plate very other Fresnel zone is
blocked The zone plate radii are approximately
given by,
18
Diffraction
Diffraction by Straight Edges Use cylindrical
waves.
19
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 N1
and 3, respectively. The first minimum occurs
when N2.
20
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
21
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 Rayleighs 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
eyes refractive index 1.333. Assume a single
surface model eye with the pupil at the surface.