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

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):

11.2.2 Two-dimensional transforms x y a

J0(u) J1(u) u

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

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

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:

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:

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 f (x) F (k) f (x) F(k) 1 1 … x k x k

A (k) f (x) k x f (x) B (k) k x f (x) A (k) x k f (x) B (k) x k

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

Point-spread function 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 y I0 (y,z) z Y Ii (Y,Z) Z Point-spread function

Example: y I0 (y,z) z Y Ii (Y,Z) Z The point-spread function is 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. y I0 (y,z) z Y Ii (Y,Z) Z

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

11.3.3 The convolution integral Convolution integral: The convolution integral of two functions f (x) and h(x) is x f (x) h (x) h (X-x) X f (x)h (x) 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:

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

Frequency convolution theorem: Please prove it. Example: Transform of a Gaussian wave packet. Transfer functions: Optical transfer function T (OTF) Modulation transfer function M (MTF) Phase transfer function F (PTF)

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

April 26,29 Fourier methods in diffraction theory Fraunhofer diffraction: dydz P(Y,Z) r R x y z Y Z X 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:

The double slit (with finite width): The single slit: z A (z) b/2 -b/2 kZ E (kZ) 2p/b 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

Apodization: Removing the secondary maximum of a diffraction pattern. Three slits: |F (kZ)|2 F (kZ) f (z) z kZ kZ -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) kZ F (kZ) z

= The Fourier transform of an individual aperture 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).

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

May 1 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

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

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

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

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

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