Fourier transform = F. Introduction to Fourier Optics, J. Goodman Fundamentals of Photonics, B. Saleh &M. Teich. x y x y x y

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1 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

2 Properties of 1D FT

3 Properties of 1D FT

4 Some frequently used functions

5 Some frequently used functions

6 Time duration and spectral width The rms width The power rms width (most of the measurement quantities) (Principles of optics 7 th Ed, , p615)

7 Time duration and spectral width

8 Widths at 1/e, 3-dB, half-maximum 1 f(t) t = 2τ.

9 Superposition of plane waves 2D Fourier transform

10 Remind!! Spatial frequency and propagation angle directional cosine : α = λν x z Λ= 1 ν x

11 Spatial frequency and propagation angle

12 Fourier and Inverse Fourier Transform ( f, f ) x y α β β α

13 Properties of 2D FT

14 Properties of 2D FT

15 FT in cylindrical (polar) coordinates ( x, y) ( r, θ ) ( f, f ) x ( ρφ, ) y In rectangular coordinate In cylindrical coordinate

16 FT in cylindrical coordinates

17 FT in cylindrical coordinates (Ex) Circular aperture : for the special case when

18

19 Special functions in Photonics

20 Special functions in Photonics

21 Special functions in Photonics

22 Fourier Transform with Lenses back focal plane Input placed against lens Input placed behind lens Input placed in front of lens

23 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) x + y x + y = Δ0 R R R1 R2 Δ 2 2

24 The Paraxial Approximation ( ) [ ] ( ) + Δ = exp exp, R R y x n jk jkn y x t l ( ) R R n f concave : 0 < f convex : 0 > f ( ) ( ) + = exp, y x f k j y x t l Phase representation of a thin lens (paraxial approximation) focal length

25 t l k ( ) ( 2 2 ) x, y = exp j x + y 2 f > 0 : convex f Types of Lenses f < 0 : concave

26 Collimating property of a convex lens Fig. 1.21, Iizuka Plane wave! z i

27 How can a convex lens perform the FT f o f o

28 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 ): 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

29 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!

30 ( ) ( ) d f d j u d k j A u U f λ υ υ + = 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,, A t d k j d f d f P d Af U + =

31 Invariance of the input location to FT

32 Imaging property of a convex lens From an input point S to the output point P ; magnification Fig. 1.22, Iizuka

33 Diffraction-limited imaging of a convex lens From a finite-sized square aperture of dimension a x a to near the output point P ;

34 Appendix : Linear systems

35 Appendix : Shift-invariant systems

36 Appendix : Linear shift-invariant causal systems

37 Example : The resonant dielectric medium p.180 Response to harmonic (monochromatic) fields : Let, Susceptibility of a resonant medium :

38 Appendix : Transfer function