fourier relations in optics near fieldfar field frequencypulse duration frequencycoherence length...
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Fourier relations in Optics
Near field Far field
Frequency Pulse duration
Frequency Coherence length
Beam waist Beam divergence
Focal plane of lens The other focal plane
Spatial dimension Angular dimension
Huygens’ Principle
E(r) E(R)
rdrR
erERE
rRkj
)()()(
dveFtf tj)()(
dtetfF tj )()(
Fourier theoremA complex function f(t) may be decomposed as a superposition integral of harmonic function of all frequencies and complex amplitude
(inverse Fourier transform)
The component with frequency has a complex amplitude F(), given by
(Fourier transform)
1)( tf )()( F
)()( ttf 1)( F
elsewhere
ttf
0
2/2/1)( )2/sin(
2)( F
elsewhere
tttf
022
)cos()( 0
]
2)[(
]2
)sin[()(
0
0
F
Useful Fourier relations in opticsbetween t and , and between x and .
2
2
)( t
etf
22
)( eF
elsewhere
Nn
nTtnTtf
0
1.....,1,022
1)(
)2
sin(
)2
sin(
2
)2
sin()(
T
NT
F
Useful Fourier relations in opticsbetween t and , and between x and .
Position or time Angle or frequency
Position or time Angle or frequency
2
2sin
)()(
0
2
2
22
0
E
j
eeEdteE
dtetfF
jj
otj
tj
2
2
2
Single- slit diffractionApplication of Fourier relation:
a
a
Single slit diffraction
X
X=0
X=a/2
= x sin
a
a
aa
k
ak
adxeE ikx)sin(
sin2
)sin2
sin()( sin
-Spatial harmonics and angles of propagation
The applications of the Fourier relation:
elsewhere
ttf
0
2/2/1)( )2/sin(
2)( F
elsewhere
axaxf
0
2/2/1)( ka
kaakF
)2/sin(2)(
?
Frequency, time, or position
N
N
i
tjtij ieEE0
)()(0
0
tindependentimet :)(
2
2*
0
2sin
2sin
1
10
t
tN
EEE
e
eeEE
tj
tjNtj
2
N
2
TimeFrequency
N
N
i
tjtij ieEE0
)()(0
0
tindependentimet :)(
2
2*
0
2sin
2sin
1
10
t
tN
EEE
e
eeEE
tj
tjNtj
2
N
2
TimeFrequency
Mode-locking
N
xx0
2
2*
sin
sin
X
XN
EEE
X
XN
AnglePosition
Diffraction grating, radio antenna array
k=2 /
1/x
k
kz
2xx
elsewhere
Nn
nTtnTtf
0
1.....,1,022
1)(
)2
sin(
)2
sin(
2
)2
sin()(
T
NT
F
(8)
The applications of the Fourier relation:
Finite number of elements
-Graded grating for focusing
-Fresnel lens
Fourier transform between two focal planes of a lens
Z=0
d fFocal plane
f(x,y) g(x,y)
A B
The use of spatial harmonics for analyses of arbitrary field pattern
yjxj yxAeyxf 22),(
Consider a two-dimensional complex electric field at z=0 given by
where the ’s are the spatial frequencies in the x and y directions.
The spatial frequencies are the inverse of the periods.
k=2 /
1/x
k
kz
2xx
yjxj yxAeyxf 22),(
zkjyjxj zyxAezyxU 22),,(
2/1222
)1
(2 yxzk
k=2 /
1/x
k
kz
2xx
Thus by decomposing a spatial distribution of electric field into spatial harmonics, each component can be treated separately.
Define a transfer function (multiplication factor) in free space for the spatial harmonics of spatial frequency x and y to travel from z=0 to z=d as
dj
yx
yx
eH2/122
2)
1(2
),(
),(
),,(22
22
yxyjxj
djkyjxj
HAe
AedyxU
yx
zyx
2/1222
)1
(2 yxzk
k=2 /
1/x
k
kz
2xx
yjxj yxAeyxU 22)0,,(
Define a transfer function (multiplication factor) in free space for the spatial harmonics of spatial frequency x and y to travel from z=0 to z=d as
dj
yx
yx
eH2/122
2)
1(2
),(
),(
),,(22
22
yxyjxj
djkyjxj
HAe
AedyxU
yx
zyx
2/1222
)1
(2 yxzk
k=2 /
1/x
k
kz
2xx
Source
z=0 z=0
E E
Nxkk x2ˆ)( 12
k1
1/x
k2
k2z
2xk1x
To generalize:
“Grating momentum”
Stationary gratings vs. Moving gratings
Deflection Deflection + Frequency shift
The small angle approximation (1/ <<)for the H function
)2
1(2)1(2)1
(22
2/122/1222
dd
dyx
)( 22
),( yxdjjkdyx eeH
A correction factor for the transfer function for the plane waves
=
D
F(x) H(x)F(x)
z=0z
xx
x defxF x 2)()0,(
D
F(x) H(x)F(x)
z=0 z
xx
xx defHxF x 2)()()(
)( 22
),( yxdjjkxyx eeH
Express F(x,z) in =x/z
xx
x defxF x 2)()0,(
D
F(x) H(x)F(x)
z=0 z
xx
xx defHxF x 2)()()(
)( 22
),( yxdjjkxyx eeH
Express F(x,z) in =x/z
The effect of lenses
A lens is to introduce a quadratic phase shift to the wavefront given by.
f
yxj
e
22
Converging lens
f
Z=0
d fFocal plane
f(x,y) g(x,y)
A B
Fourier transform using a lens
yxyxj
yx ddeFyxf yx )(2),(),(
f
yyxxj
yx
yxjyx
djf
yxj
jkdBB
eA
eFeeezyxU yxyx
2
02
0
22
22
)()(
)(2)(
),(
),(),,(
Z=0
d fFocal plane
f(x,y) g(x,y)
A B
yxyxj
yx ddeFyxf yx )(2),(),(
),(),( ))(( 22
yxfdjjkd
yx FeeA yx
),(),(f
y
f
xFhyxg f
Z=0
f fFocal plane
f(x,y) g(x,y)
),()/(f
y
f
xFfjhl
Huygens’ Principle
E(r) E(R)
rdrR
erERE
rRkj
)()()(
:
Recording of full information of an optical image, including the amplitude and phase.
Holography
Amplitude only: 2
* EEE
Amplitude and phase
)(cos2
))(())((22
)()(
xEEEE
EexEEexE
RR
Rxj
Rxi
k1k2
)2
sin(2
d
A simple example of recording and reconstruction:
A simple example of recording and reconstruction:
k1k2
)2
sin(2
d
)2
sin(2
d
)2
sin(2
d
/2
?
k2k1
k1k2
)2
sin(2
d
Another example: Volume hologram
)2
sin(2
d
Volume grating
)2
sin(2
d
k1
)2
sin(2
d
k1
k1 k2
d
D
C
B
A
cos2dCDCBAB
Bragg condition
d
D
C
B
A
Another example:Image reconstruction of a point illuminated by a plane wave.
Writing
Reading
),(),(),(),( 222yxEeyxEEeyxEEEeyxEE xjk
rxjk
rrxjk
rxxx
Er
E(x,y)
Recorded pattern
jkzzjkxjkr
zjkxjkr
jkzr
diffracted
eyxEeyxEEeyxEEeE
E
zxzx
),(),(),( 22
),(),(),(),( 222yxEeyxEEeyxEEEeyxEE xjk
rxjk
rrxjk
rxxx
Recorded pattern
Diffracted beam when illuminated by ER
