1 / 30

11.2 Fourier transforms 11.2.1 One-dimensional transforms

Chapter 11 Fourier optics. April 15,17 Fourier transform. 11.2 Fourier transforms 11.2.1 One-dimensional transforms. Complex exponential representation :. In time domain:. Note: Other alternative definitions exist. Notations:. Proof:.  Fourier transform   Inverse Fourier transform .

abena
Download Presentation

11.2 Fourier transforms 11.2.1 One-dimensional transforms

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 11 Fourier optics April 15,17 Fourier transform 11.2 Fourier transforms 11.2.1 One-dimensional transforms Complex exponential representation: In time domain: Note: Other alternative definitions exist. Notations: Proof: Fourier transform Inverse Fourier transform The fancy “F” (script MT) is hard to type.

  2. Example:Fourier transform of the Gaussian function 1) The Fourier transform of a Gaussian function is again a Gaussian function (self Fourier transform functions). 2) Standard deviations (where f (x) drops to e-1/2):

  3. 11.2.2 Two-dimensional transforms x a y

  4. J0(u) J1(u) u

  5. Read: Ch11: 1-2 Homework: Ch11: 2,3 Due: April 26

  6. d (x) 1 x 0 d (x- x0) 1 x x0 0 April 19 Dirac delta function 11. 2.3 The Dirac delta function Dirac delta function: A sharp distribution function that satisfies: A way to show delta function Sifting property of the Dirac delta function: If we shift the origin, then

  7. Delta sequence: A sequence that approaches the delta function when the distribution is gradually narrowed. The 2D delta function: The Fourier representation of delta function:

  8. Displacement and phase shift: The Fourier transform of a function displaced in space is the transform of the undisplaced function multiplied by a linear phase factor. Proof:

  9. Fourier transform of some functions: (constants, delta functions, combs, sines and cosines): f(x) F(k) f(x) F(k) 1 2p … x k x k 0 0 0 0 f(x) F(k) f(x) F(k) 1 1 … x k x 0 0 k 0 0

  10. A(k) f(x) k x 0 f(x) B(k) k x 0 f(x) A(k) x k 0 f(x) B(k) k x 0

  11. Read: Ch11: 2 Homework: Ch11: 4,8,10,11,12,17 Due: April 26

  12. Z z Ii(Y,Z) I0(y,z) Y y April 22, 24 Convolution theorem 11.3 Optical applications 11.3.2 Linear systems Linear system: Suppose an object f (y, z) passing through an optical system results in an image g(Y, Z), the system is linear if 1) af (y, z) ag(Y, Z), 2) af1(y, z) + bf2(y, z)  ag1(Y, Z) + bg2(Y, Z). We now consider the case of 1) incoherent light (intensity addible), and 2) MT = +1. The flux density arriving at the image point (Y, Z) from dydz is Point-spread function

  13. z Z Ii(Y,Z) I0(y,z) y Y z Z I0(y,z) Ii(Y,Z) y Y Example: • The point-spread functionis the irradiance produced by the system with an input point source. In the diffraction-limited case with no aberration, the point-spread function is the Airy distribution function. • The image is the superposition of the point-spread function, weighted by the source radiant fluxes.

  14. Space invariance: Shifting the object will only cause the shift of the image:

  15. f(x) x h(x) x h(X-x) x X f(x)h(x) X 11.3.3 The convolution integral Convolution integral: The convolution integral of two functionsf (x) and h(x) is Symbol: g(X) = f(x)h(x) Example 1: The convolution of a triangular function and a narrow Gaussian function. Question: What is f(x-a)h(x-b)? Answer:

  16. f(x) x h(x) x h(X-x) x X f(x)h(x) X Example 2: The convolution of two square functions. The convolution theorem: Proof: Example:f (x) and h(x) are square functions.

  17. Frequency convolution theorem: Please prove it. Example: Transform of a Gaussian wave packet. Transfer functions: Optical transfer functionT (OTF) Modulation transfer functionM (MTF) Phase transfer functionF (PTF)

  18. Read: Ch11: 3 Homework: Ch11: 18,24,27,28,29,34,35 Due: May 3

  19. Y y Z P(Y,Z) r dydz R Y x X z Z April 26,29 Fourier methods in diffraction theory 11.3.4 Fourier methods in diffraction theory Fraunhofer diffraction: Aperture function: The field distribution over the aperture: A(y, z) = A0(y, z) exp[if (y, z)] • Each image point corresponds to a spatial frequency. • The field distribution of the Fraunhofer diffraction pattern is the Fourier transform of the aperture function:

  20. A(z) E(kZ) z kZ b/2 -b/2 2p/b The single slit: Rectangular aperture: Fraunhofer-: The light interferes destructively here. Fourier-: The source has no spatial frequency here. The double slit (with finite width): f(z) h(z) g(z)  = z z z b/2 -b/2 a/2 -a/2 a/2 -a/2 F(kZ) H(kZ) G(kZ) × = kZ kZ kZ

  21. F(kZ) kZ Three slits: |F(kZ)|2 F(kZ) f(z) kZ kZ z 0 a -a Apodization: Removing the secondary maximum of a diffraction pattern. • Rectangular aperture  sinc function  secondary maxima. • Circular aperture  Bessel function (Airy pattern)  secondary maxima. • Gaussian aperture  Gaussian function  no secondary maxima. f(z) z

  22. Array theorem: The Fraunhofer diffraction pattern from an array of identical apertures = The Fourier transform of an individual aperture × The Fourier transform of a set of point sources arrayed in the same manner. Convolution theorem z z  = y y y Example: The double slit (with finite width).

  23. Read: Ch11: 3 Homework: Ch11: 37,38,40 Due: May 3

  24. May 1 Spectra and correlation 11.3.4 Spectra and correlation Considering a laser pulse described by E(t) = f (t). The temporal radiant flux is The total energy is Parseval(1755-1836, French mathematician)’s formula: If F(w) =F{f(t)}, then |F(w)|2 is the power spectrum. Unitarity of Fourier transform

  25. f (t) t Application: Lorentzian profile: g w w0

  26. Nature line width:The frequency bandwidth caused by the finite lifetime of the excited states. Line broadening mechanisms: • Natural broadening: Governed by the uncertainty principle. Lorentzian profile. • Doppler broadening: The light emitted will be red or blue shifted depending on the velocity of the emitting atoms relative to the observer. Gaussian profile. • Pressure broadening: The collision with other atoms interrupts the emission process. Lorentzian profile. Autocorrelation: The autocorrelation of f (t) is Wiener-Khintchine theorem: Determining the spectrum by autocorrelation: Symbol: cff (t) = f (t)f (t). Prove: Let h(t) = f *(-t), then

  27. Cross Correlation: The cross correlation of f (t) and h(t) is Symbol: cfh(t) = f(t)h(t). For real functions, Properties (please prove them): 1) f (t)h(t) = f *(-t)h(t). Can be treated as the definition of cross correlation. 2) If f is an even function, then f (t)h(t) = f (t)h(t). 3) Cross-correlation theorem: F{f(t)h(t)} = F{f (t)}* ·F{h(t)} Applications of cross correlation: Optical pattern recognition Optical character recognition Rotational fitting in laser spectroscopy

  28. Read: Ch11: 3 Homework: Ch11: 39,49,50 Due: May 8

  29. All the world’s a stage, and all the men and women merely players. William Shakespeare

More Related