Plane waves and spatial frequency Plane waves and spatial frequency
A plane wave
Complex representation Complex representation
[ ]
1 cos(2 ) cos( )
2 A B ωt α β α β
= + + + −
( , ) cos( )
ocos( )
E z t = E ω t − kz = E ω t − kz E z t ( , ) = E e
j(ωt kz− )= E e
o j(ωt kz− ){ }
Re a t b t( ) ( ) = A B cos(2ω α βt+ + )
Complex representation Complex representation
( , ) cos( )
ocos( )
E z t = E ω t − kz = E ω t − kz E z t ( , ) = E e
j(ωt kz− )= E e
o j(ωt kz− )Now, it’s identical !!
[ ] [ ] 1 * 1
( ) ( ) Re ( ) Re ( ) Re cos( )
2 2
a t b t = a t b t = ⎡⎣AB ⎤⎦ = AB α β− (real form)
(complex form)
(
a+a*)(
b b+ *)
= ab+ab*+a b* +a b* *Keep the complex representation until you reach final answer !!!
Consider the time-averaged values which are meaningful, rather than the instantaneous values of many physical quantities.
(Since the field vectors are rapidly varying function of time; for example λ = 1 μm has 0.33 x 10-14sec time-varying period!)
Complex representation of real quantities : Examples
Complex representation of real quantities : Examples
Complex representation of real quantities : Examples
Complex representation of real quantities : Examples
Plane waves : 2D Plane waves : 2D
⎥⎦⎤
⎢⎣⎡ =
=
= − − − ⋅ e
k c e
E e
y x E t y x
EG( , , ) Gr( , ) jωt G0(0,0) j(ωt kGrG) ; G ω ˆ
Spatial frequency Spatial frequency
2 2
cos sin 2
x2
yk i j
k f i f j
π θ π θ
λ λ
π π
= +
= +
G
G
θ
e
Plane waves : 3D Plane waves : 3D
x
y z
e e
cos 1
a= − α
cos 1
b= − β cos 1
c = − γ
(α, β, γ)… directional cosine
x y z
f f f
α λ = β λ = γ λ =
3D Plane waves : Example 1.2
3D Plane waves : Example 1.2
Physical meaning of spatial frequency Physical meaning of spatial frequency
cos sin
= sin
y y y
f f θ φ f
β λ φ λ
λ λ
= → = → =
φ θ
spherical parabolic planar
Spatial frequency and propagation angle Spatial frequency and propagation angle
z
directional cosine : α λν =
x1 ν
xΛ =
Fourier transform and Diffraction Fourier transform and Diffraction
Spherical wave from source Po
Huygens’ Secondary wavelets on the wavefront surface S
Obliquity factor: unity at C where χ=0, zero at high enough zone index ( Remind!! )
{ }
1 exp(iks) /s iλ
⇒ The field at P from a point source with an infinitesimal area at (xo, yo),
Diffraction under paraxial approx.
Diffraction under paraxial approx.
Huygens-Fresnel principle
“Every unobstructed point of a wavefront, at a given instant in time, serves as a source of secondary wavelets (with the same frequency as that of the primary wave).
The amplitude of the optical field at any point beyond is the superposition of all these wavelets (considering their amplitude and relative phase).”
Huygens’s principle:
By itself, it is unable to account for the details of the diffraction process.
It is indeed independent of any wavelength consideration.
Fresnel’s addition of the concept of interference
Again, remind Huygens and Fresnel ……..
After the Huygens-Fresnel principle ……
Fresnel’s shortcomings :
He did not mention the existence of backward secondary wavelets,
however, there also would be a reverse wave traveling back toward the source.
He introduce a quantity of the obliquity factor, but he did little more than conjecture about this kind.
Arnold Johannes Wilhelm Sommerfeld : Rayleigh-Sommerfeld diffraction theory A very rigorous solution of partial differential wave equation.
The first solution utilizing the electromagnetic theory of light.
Gustav Kirchhoff : Fresnel-Kirhhoff diffraction theory
A more rigorous theory based directly on the solution of the differential wave equation.
He, although a contemporary of Maxwell, employed the older elastic-solid theory of light.
He found K(χ) = (1 + cosθ )/2. K(0) = 1 in the forward direction, K(π) = 0 with the back wave.
Fraunhofer diffraction and Fourier transform
Fraunhofer diffraction and Fourier transform
Fresnel diffraction and Fourier transform Fresnel diffraction and Fourier transform
Fourier optics
Fourier optics
Fourier optics
Fresnel diffraction and convolution Fresnel diffraction and convolution
PSF means
Impulse response function of free space in Fresnel approximation
Impulse response function of free space in Fresnel approximation
zi = d, in general,
h(x,y)
Therefore, free-space propagation can be treated as a convolution in the Fresnel approximation!
Impulse response function and transfer function Impulse response function and transfer function
FT
PSF (or, Impulse Response function)
“Transfer function”
< proof >
Appendix : Transfer function
Appendix : Transfer function
Huygens’ wave front construction
Given wave
Given wave--front at tfront at t
Allow wavelets to evolve for time Δt
r = c Δt ≈ λ
New wavefront
What about –r direction?
(π-phase delay when the secondary wavelets, Hecht, 3.5.2, 3nd Ed)
Construct the wave front tangent to the wavelets Every point on a wave front is a source of secondary wavelets.
i.e. particles in a medium excited by electric field (E) re-radiate in all directions i.e. in vacuum, E, B fields associated with wave act as sources of additional fields
secondary wavelets
Secondary wavelet