# Plane waves and spatial frequencyPlane waves and spatial â€؛ ~shsong â€؛ 2-spatial آ ...

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Plane waves and spatial frequencyPlane waves and spatial frequency

A plane wave

Complex representationComplex representation

[ ]1 cos(2 ) cos( ) 2

A B tω α β α β= + + + −

( , ) cos( ) cos( )oE z t E t kz E t kzω ω= − = − ( ) ( )( , ) j t kz j t kzoE z t E e E e ω ω− −= =

{ }Re ( ) ( ) cos(2 )a t b t A B tω α β= + +

Complex representationComplex representation ( , ) cos( ) cos( )oE z t E t kz E t kzω ω= − = −

( ) ( )( , ) j t kz j t kzoE z t E e E e ω ω− −= =

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 10-14 sec time-varying period!)

Complex representation of real quantities : ExamplesComplex representation of real quantities : Examples

Complex representation of real quantities : ExamplesComplex representation of real quantities : Examples

Plane waves : 2DPlane waves : 2D

⎥⎦ ⎤

⎢⎣ ⎡ === ⋅−−− e

c keEeyxEtyxE rktjtjr ˆ ; )0,0(),(),,(

)( 0

ωωω

Spatial frequencySpatial frequency

2 2cos sin

2 2 x y

k i j

k f i f j

π πθ θ λ λ π π

= +

= + θ e

Plane waves : 3DPlane waves : 3D

x

y

z

ee

1cosa α−=

1cosb β−=

1cosc γ−=

(α, β, γ)… directional cosine

x y zf f fα λ β λ γ λ= = =

3D Plane waves : Example 1.23D Plane waves : Example 1.2

Physical meaning of spatial frequencyPhysical meaning of spatial frequency

cos sin = sin y y yf f f θ φβ λ φ λ

λ λ = → = → =

φ

θ

spherical parabolic planar

Spatial frequency and propagation angleSpatial frequency and propagation angle

z

directional cosine : xα λν=

1

xν Λ =

Fourier transform and DiffractionFourier 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 opticsFourier optics

Fresnel diffraction and convolutionFresnel 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 functionImpulse response function and transfer function

FT

PSF (or, Impulse Response function) “Transfer function” < proof >

Appendix : Transfer functionAppendix : Transfer function

Huygens’ wave front construction

Given waveGiven 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

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