Fourier transform Introduction to Fourier Optics, J. Goodman Fundamentals of Photonics, B. Saleh &M. Teich f( x, y) FT g( f, f ) f( x, y) IFT g( f, f ) x x y y + 1 { g( f, ) } x fy { π } f( x, y) = g( f, f )exp j2 ( xf + yf ) df df =F { f( x, y) } { π } + x y x y g( f, f ) = f( x, y)exp j2 ( f x+ f y) dxdy = F x y x y x y
Properties of 1D FT
Properties of 1D FT
Some frequently used functions
Some frequently used functions
Time duration and spectral width The rms width The power rms width (most of the measurement quantities) (Principles of optics 7 th Ed, 10.8.3, p615)
Time duration and spectral width
Widths at 1/e, 3-dB, half-maximum 1 f(t) t = 2τ.
Superposition of plane waves 2D Fourier transform
Properties of 2D FT
Properties of 2D FT
Properties of 2D FT
Fourier and Inverse Fourier Transform ( f, f ) x y α β β α
Fourier Transform with Lenses back focal plane Input placed against lens Input placed behind lens Input placed in front of lens
A thin lens as a phase transformation Intro. to Fourier Optics, Chapter 5, Goodman. φ ( x, y) = knδ( x, y) + k[ Δ0 Δ( x, y) ] ( x, y) = exp[ jkδ 0 ] exp[ jk( n 1) Δ( x y) ] t l, U ( x, y) = t ( x, y) U ( x y) ' l l l, Ul ( x, y) U ' l ( x, y) R 1 >0 (concave) R 2 <0 (convex) ( x, y) 2 2 2 x + y x + y = Δ0 R1 1 1 + R2 1 1 2 R1 R2 Δ 2 2
The Paraxial Approximation ( ) [ ] ( ) + Δ = 2 1 2 2 0 1 1 2 1 exp exp, R R y x n jk jkn y x t l ( ) 2 1 1 1 1 1 R R n f concave : 0 < f convex : 0 > f ( ) ( ) + = 2 2 2 exp, y x f k j y x t l Phase representation of a thin lens (paraxial approximation) focal length
t l k ( ) ( 2 2 ) x, y = exp j x + y 2 f > 0 : convex f Types of Lenses f < 0 : concave
Collimating property of a convex lens Fig. 1.21, Iizuka Plane wave! z i
How can a convex lens perform the FT f o f o
Fourier transforming property of a convex lens The input placed directly against the lens Pupil function ; P ( x, y) 1 inside the lens aperture = 0 otherw ise k U x y U x y P x y j x y 2 f (, ) = l (, ) (, ) exp ( + ) ' 2 2 l From the Fresnel diffraction formula ( z = f ): U l U l k 2 2 exp j ( u + υ ) 2 f ' k 2 2 2π U f ( u, υ ) = Ul ( x, y) exp j ( x + y ) exp j ( xu + yυ ) dxdy jλ f 2 f λ f k 2 2 exp j ( u + υ ) 2 f 2π U f ( u, υ) = U l ( x, y) P ( x, y) exp j ( xu + yυ) dxdy jλ f λ f Quadratic phase factor Fourier transform
Fourier transforming property of a convex lens The input placed in front of the lens k d 2 2 Aexp j 1 ( u ) 2 f f + υ 2π U f ( u, υ) = Ul( x, y) exp j ( xu yυ) dxdy jλf + λf If d = f A 2π U f ( u, υ) = Ul( x, y) exp j ( xu yυ) dxdy jλf + λf Exact Fourier transform!
( ) ( ) d f d j u d k j A u U f λ υ υ + = 2 2 2 exp, ( ) ( ) η ξ υη ξ λ π η ξ η ξ d d u d j d f d f P t A + 2 exp,, Fourier transforming property of a convex lens Fourier transforming property of a convex lens The input placed behind the lens Scaleable Fourier transform! By decreasing d, the scale of the transform is made smaller. ( ) ( ) ( ) η ξ η ξ η ξ η ξ, 2 exp,, 2 2 0 A t d k j d f d f P d Af U + =
Invariance of the input location to FT
Imaging property of a convex lens From an input point S to the output point P ; magnification Fig. 1.22, Iizuka
Diffraction-limited imaging of a convex lens From a finite-sized square aperture of dimension a x a to near the output point P ;
FT in cylindrical (polar) coordinates ( x, y) ( r, θ ) ( f, f ) x ( ρφ, ) y In rectangular coordinate In cylindrical coordinate
FT in cylindrical coordinates
FT in cylindrical coordinates (Ex) Circular aperture : for the special case when
Special functions in Photonics
Special functions in Photonics
Special functions in Photonics
Appendix : Linear systems
Appendix : Shift-invariant systems
Appendix : Linear shift-invariant causal systems
Example : The resonant dielectric medium p.180 Response to harmonic (monochromatic) fields : Let, Susceptibility of a resonant medium :
Appendix : Transfer function