Plane waves and spatial frequency A plane wave
Complex representation E(,) z t = E cos( ωt kz) = E cos( ωt kz) o Ezt (,) = Ee = Ee j( ωt kz) j( ωt kz) o = 1 2 A B t + + + [ cos(2 ω α β ) cos( α β )] { } Re atbt () () = ABcos(2 ωt+ α + β )
Complex representation E(,) z t = E cos( ωt kz) = E cos( ωt kz) o Ezt (,) = Ee = Ee j( ωt kz) j( ωt kz) o 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!) (real form) (complex form) 1 * 1 atbt () () = Re [ at ()] Re [ bt ()] = Re AB ABcos( α β ) 2 = 2 ( )( ) a + a * b + b* = ab + ab* + a * b + a * b* Now, it s identical!! Keep the complex representation until you reach final answer!!!
Complex representation of of real quantities :: Examples
Complex representation of of real quantities :: Examples
Plane waves : 2D Plane waves : 2D = = = e c k e E e y x E t y x E r k t j t j r ˆ ; (0,0) ), ( ),, ( ) ( 0 ω ω ω
Spatial frequency 2π 2π k = cos θ i + sin θ j λ λ k = 2 π f i + 2 π f j x y θ e
Plane waves : 3D (α, β, γ) directional cosine z c = cos 1 γ e b = cos y 1 β x a = cos 1 α α = λf β = λ f γ = λf x y z
3D Plane waves : Example 1.2
Physical meaning of spatial frequency cosθ sinφ β = λ f f = = sin φ = λ f λ λ y y y θ φ spherical parabolic planar
Spatial frequency and propagation angle directional cosine : α = λν x z Λ= 1 ν x
Fourier transform and Diffraction The field at P from a point source with an infinitesimal area at (x o, y o ), ( Remind!! ) 1 exp( ) / iλ { iks s } Spherical wave from source P o Obliquity factor: unity at C where χ=0, zero at high enough zone index Huygens Secondary wavelets on the wavefront surface S
Diffraction under paraxial approx.
Again, remind Huygens and Fresnel.. 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
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. 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. 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.
Fraunhofer diffraction and Fourier transform
Fresnel diffraction and Fourier transform Fourier optics
Fourier optics
PSF means Fresnel diffraction and convolution
Impulse response function of free space in Fresnel approximation z i = 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 FT PSF (or, Impulse Response function) < proof > Transfer function
Appendix : Transfer function
Huygens wave front construction 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 New wavefront Construct the wave front tangent to the wavelets Secondary wavelet r = c Δt λ secondary wavelets Given wave-front at t Allow wavelets to evolve for time Δt What about r direction? (π-phase delay when the secondary wavelets, Hecht, 3.5.2, 3nd Ed)