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3. Optics (2) Diffraction Grating
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Basic principles: Diffraction
Basic principles: Fraunhofer and Fresnel diffraction
Single- & Multi-slit
The grating equation again
Resolving power
Blazed gratings
• Consider the electromagnetic wave E and H.
We define them as
y = exp(iwt ) f ( x, y,z)
or
where
Thus
f ( x, y,z) = f 0 exp[i( k1x + k2 y + k3z)]
2
æ
2
2ö
w
k = ç k1,k2 ,
2 - k1 - k2 ÷
c
è
ø
for a plane wave
(because
k12 + k22 + k32 = w
2
c2
)
éæ
ù
2
2
2 ö
w
f ( x, y,z) = f 0 expêiç k1 x + k2 y +
2 - k1 - k 2 z÷ú
c
øû
ëè
In general, the above equation can be written as
éæ
ù
2
2
2 ö
w
f ( x, y,z) = òò A( k1,k2 ) expêiç k1 x + k2 y +
2 - k1 - k 2 z÷údk1dk 2
c
øû
ëè
At z=0
f ( x, y,0) =
òò A(k ,k ) exp[i(k x + k y)]dk dk
1
2
1
2
1
2
Applying Fourier inversion transform for the above equation,
A( k1,k2 ) =
¥ ¥
1
(2p )
2
ò ò exp[i(k x + k y)] f ( x, y,0) dxdy
1
2
-¥ -¥
However, f=0 outside the hole R at z=0. Then we have
A( k1,k2 ) =
1
(2p )
2
òò exp[i(k x + k y)] f ( x, y,0) dxdy
1
2
R
x
k
R
y
z
Fraunhofer vs. Fresnel diffraction
• In Fraunhofer diffraction, both
incident and diffracted waves are
considered to be plane (i.e. both S
and P are at large distance)
• If either S or P are close enough
that wavefront curvature is not
negligible, then we have Fresnel
diffraction
S
P
Fraunhofer diffraction by very long slit
First, let's assume the rectangle slit (left figure). The length and the
width of the slit are assumed to be 2b and 2a, respectively. propagate
along z-axis
The incident plane wave is described as y = f ( z ) expéë-i wt ùû, f ( z ) = Aexpéëikzùû
f=A when z=0. Therefore, from A( k1,k2 ) =
A( k1,k2 ) =
x
A
(2p )
2
z
2a
y
2b
1
(2p )
2
òò exp[i(k x + k y)] f ( x, y,0) dxdy
1
R
òò exp[-i( k x + k y )]dxdy
1
R
2
2
A( k1,k2 ) =
A
(2p )
2
òò exp[-i( k x + k y )]dxdy
1
2
R
After the passage through the slit (z>0)
Assuming k1=k and k2=k
(
)
A k1 ,k2 =
A
b
é
ù
é
ù
exp
-ik
a
x
dx
ò -a ë
û ò -b exp ë-ikb y û dy
a
( )
2
2p
A exp[ikaa] - exp[-ikaa] exp[ikbb] - exp[-ikbb]
=
2
ika
ikb
(2p )
4 A sin[ kaa] sin[ kbb]
=
2
ka
kb
(2p )
x
x
k

z
2a
y
2b
2a
y
z
The intensity ratio of A(k1,k2) is therefore,
A( k1,k2 )
A(0,0)
2
2
ìï 4 A sin[ kaa] sin[ kbb] üï 2
=í
ý
ïî (2p ) 2
ka
kb ïþ
2
2
ì sin[ kaa] ü ì sin[ kbb] ü
=í
ý í
ý
k
a
a
k
b
b
î
þ î
þ
ìï 4 A sin[ kaa] sin[ kbb] üï 2
í
ý
ïî (2p ) 2
ka
kb ïþ
a
b
When b>>1 (long-slit), 0. Accordingly,
2
ìï sin[ kasin(q )] üï
ì sin[ kaa] ü
\ I(q ) = í
ý I0 = í
ý I0
ï
î kaa þ
î kasin(q ) ïþ
2
Or supposing =ka sin()=2a/sin()
2
ì sinB ü
\I q = í
ý I0
î B þ
x
()
x
k

z
2a
y
2b
2a
y
z
Single Slit Fraunhofer diffraction:
Effect of slit width
• Minima for sin [k a sin] = 0
• k a sin = p
• First minima at sin  = /2a

k a sin
2a
Single Slit Fraunhofer diffraction:
Effect of slit width
•
•
•
•
Width of central max = /a
This relation is characteristic of all Fraunhofer diffraction
If a is very large  0 and a point source is imaged as a point
If a is very small (~) /2 and light spreads out across screen
Similary, you can consider Fraunhofer diffraction from a circular aperture
r
• Consider a circular hole (radius of r) as shown above.
Fraunhofer vs. Fresnel diffraction (cont.)

’
r’
S

r
h’
d’
h
d
P
Now calculate variation in (r+r’) in going from one side of aperture to the other.
Call it 
(
)
2
(
)
2
D = d' + h'+ d + d + h+ d - d' 2 + h' 2 - d 2 + h2
2
2
Because (h+)/d<<1 and (h’+)/d’<<1
2ö
2ö
2ö
2ö
æ
æ
æ
æ
æ
ö
æ
ö
æ
ö
æ
ö
1 h'+ d ÷ ç 1 h+ d ÷
1 h' ÷ ç 1 h ÷
ç
D @ d' ç 1 + ç
+
d
1
+
d'
1
+
÷
ç
÷
ç ÷ - d 1+ ç ÷
ç 2 è d' ø ÷ ç 2 è d ø ÷
ç 2 è d' ø ÷ ç 2 è d ø ÷
è
ø è
ø
è
ø è
ø
æ h' h ö
1æ 1 1 ö 2
D = ç + ÷ d + ç + ÷d +
è d' d ø 2 è d d' ø

’
r’
S
d’

h’
r
h
d
P
(cont.)
æ h' h ö 1 æ 1 1 ö 2
D = ç + ÷ d + ç + ÷d +
è d' d ø 2 è d d' ø

By the way, since ~0 and ’~0,
’
r’
sin’≈ h’/d’ and sin ≈ h/d
S
d’

h
’
h
d
sin’ + sin =  ( h’/d + h/d )
First term = path difference
Thus the second term is measure of curvature of wavefront.
Fraunhofer Diffraction is defined when the curvature of the
wavefront is small enough, that is,
1æ 1 1ö 2
ç + ÷d áá l
2 è d'
r
dø
P
• If aperture is a square, the area is 2.
• Then we have the Fraunhofer diffraction if,
d2
d>
l
or ,
d>
area of
aperture
l
Fraunhofer or far field limit
Diffraction from two slits:
• Now, let’s suppose the collimated beam falls on
two slits.
• Then we project a pattern at right: a series of
bright interference fringes modulated by the
envelope of the single slit.
Note that, so far, we assumed the slit width = 2a
Diffraction from two slits:
• Mathematically, we can show that the bright
fringes are regularly spaced, and modulated by the
single slit envelope:
Same as for single slit (the envelope)
æ sin b ö
2
IP = I o ç
cos
a
÷
è b ø
2
Interference term
b
b = k sin q
2
a
a = k sin q
2
Diffraction from an array of N slits:
… separated by a distance a and of width b
y=(N-1)a + b

y=(N-1)a
y=3a+b
y=3a
y=2a+b
y=2a
y=a+b
y=a
y=b
y=0




P
Diffraction from an array of N slits
• It can be shown that,
Same as for single slit (the envelope)
æ sin b
I P = I o çç
è b
ö
÷÷
ø
• where,
2
æ sin Na ö
ç
÷
è sin a ø
2
Interference term
b
b = k sin q
2
and
Slit width
a
a = k sin q
2
Slit interval
Diffraction and interference for N slits
The diffraction term
• Minima for sin  = 0
•  = ±p = k(b/2)sin 
or sin = ±p(/b)
k=2/
æ sin b ö
ç
÷
è b ø
2
• the same as single slit
because
æ sinNa ö2
ç
÷
è sina ø
The interference term
• Amplitude due to N coherent sources
• It can see this by adding N phasors that are 2
• Double slit when N=2
out of phase (see next page).
• Single slit when N=1
Diffraction and interference for N
slits:
Interference term
æ sin b
I P = I o çç
è b
•
•
•
•
2
ö æ sin Na ö
÷÷ ç
÷
sin
a
ø
ø è
2
Maxima occur at  = ±m (m = 0,1, 2, 3,..)
Thus maxima occur at sin  = ±m/a
This is the same result we have derived for Young’s double slit
Intensity of principal maxima, I = N2Io
Diffraction and interference for N
slits:
Interference term (cont.)
æ sin b
I P = I o çç
è b
2
ö æ sin Na ö
÷÷ ç
÷
sin
a
ø
ø è
2
• Minima occur for  = /N, 2/N, … (N-1)/N
Thus principal maxima have a width determined by zeros on
each side
• Since  = (/)a sin  = ±/N ,
the angular width is determined by
sin  = /(Na)
• Thus peaks are N times narrower than in a single slit pattern
(also a > b).
Diffraction gratings
• As we learned, it is composed of systems with
many slits per unit length – usually about
1000/mm
• Also it is usually used in reflection
• Thus principal maxima are vary sharp
• The width of peaks  = 2/N
• As N gets large the peak gets very narrow
Diffraction limit and Airy disks
• The diffraction pattern resulting from a uniformly-illuminated
circular aperture has a bright region in the center (Airy disk)
together with the series of concentric bright rings (Airy pattern).
• The angle at which the first minimum occurs, measured from the
direction of incoming light, is given by the approximate formula:
sin = 1.22/D  
where θ is in radians and λ is the wavelength of the light and D is
the diameter of the aperture.
f
D


Fraunhofer diffraction and spatial resolution
• Suppose two point sources or objects are far away (e.g. two
stars)
• Imaged with some optical system
• Two Airy patterns
– If S1, S2 are too close together the Airy patterns will overlap
and become indistinguishable
S1
S2

Fraunhofer diffraction and spatial resolution
• Assume S1, S2 can just be resolved when maximum of one
pattern just falls on minimum (first) of the other
• Then the angular separation at lens,
e.g. telescope D = 1 m and  = 500  10-9 m
fmin
1.22 l 1.22 ´ 5 ×10-7
=
=
= 6.6 ´10-7 rad = 0.14"
D
1.0
e.g. human eye D ~ 7mm
fmin
1.22 ´ 5 ×10-7
-5
»
=
8.7
´10
rad = 18"
-3
7 ´10
 Comparable to the apparent size of Saturn
Grism
• A grism is a combination of a prism and grating arranged so that
light at a chosen central wavelength passes straight through.
• The advantage of a grism is that it can be placed in a filter wheel
and treated like another filter. The basic relationships required to
design a grism are
mlT = ( n -1) sin f
where  is the central wavelength; T is
the number of lines per millimeter of the
grating; n is the refractive index of the
prism material;  is the prism apex angle.
The Fourier Transform Spectrometer (FTS)
• The FTS is a scanning interferometer
with collimated light as a input.
• A typical scheme is shown below.
For a collimated monochromatic
beam, the intensity at the detector is
determined by the phase difference
2(xa+xb).
• The primary disadvantage for
astronomical work is the fact that the
measurements require a time
sequence to determine the spectrum.
Fabry-Perot Interferometer
• The Fabry-Perot interferometer is an imaging spectrometer
which is formed by placing a device called an "etalon" in
the collimated beam.
• The etalon consists of two plane- parallel plates with thin,
highly reflective coatings on their inner faces. The plates are
in near contact but separated by a distance l.
• Assuming that the refractive index of the medium in the gap
is n (usually n=1) and  is the angle of incidence of a ray on
the etalon (usually very small), then multiple reflections and
interference within the gap occurs and the wavelengths
transmitted with maximum intensity obey the relation:
m=2nlcos