Fourier transform and optics

ft: Fourier transform
go: geometric optics
wo: wave optics

Fourier transform

Main reference

• Lectures from Prof. Brad Osgood (Stanford)

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• Analysis and synthesis of period function
• Fourier series of function with period 1
• Inner product between functions
• Complex exponential as orthonormal basis for functions

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• Fourier series of function with period T
• Setup for Fourier transform as Fourier series with T goes to infinity
• Variables of Fourier and inverse Fourier transform

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• Shifting property
• Scaling property
• Derivative property
• Duality with reversed function

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• Convolution
• Convolution in one domain is multiplication in the other domain
• Convolution is commutative

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• Problem with classic Fourier transform definition
• Rapidly decreasing functions that work for classic definition
• Test function and distribution
• Delta function as distribution
• Ordinary function as distribution
• Fourier transform in context of distribution
• Some examples

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• Sampling property of delta function via multiplication
• Shifting property of delta function via convolution
• Scaling of delta function

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• Setup for Shah function
• Definition
• Poisson summation formula and Fourier transform of Shah function

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• Sampling theorem for bandlimited function
• Aliasing with undersampling
• Example of aliasing

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• Setup for discrete Fourier transform (DFT)
• Sampled form of function in time domain
• Continuous Fourier transform of sampled form of function
• Sampling in frequency domain to get sampled form of Fourier transform (of sampled form of function)
• Replace sampled forms with discrete functions
• Define DFT

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• Discrete complex exponentials
• Orthogonality between discrete complex exponentials
• Define inverse DFT
• Some examples of DFT
• Periodicity induced by discrete complex exponentials
• Duality for reversed function
• Shifting property in DFT

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• Discrete version of convolution
• Commutative and shifting property in convolution

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• Discrete version of delta function
• Multiplication and convolution with discrete delta function

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• Definition of linear system
• Sampling is a linear system
• Integration with kernel as infinite dimensional matrix multiplication
• Integration with kernel is the only form of continuous linear system
• Delta function and impulse response
• Impulse response of Fourier transform

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• Linear system in the case of convolution
• Delay operator
• Convolution with a delayed function is delay of (convolution with the function)
• Linear time-invariant system (LTI)
• Convolution is the only form of LTI system

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• Fourier transform of LTI system
• Complex exponentials are eigenfunctions of LTI system

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• Discrete LTI system with matrix-vector multiplication
• Complex exponentials are eigenvectors of discrete LTI system

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• 2D Fourier transform
• Spatial frequency

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• High dimensional Fourier transform of separable functions

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• Shifting property of high dimensional Fourier transform
• General scaling property of high dimensional Fourier transform
• Properties of high dimensional delta function

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• 2D Shah function and its Fourier transform

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• Example: Fraunhofer diffraction
• Setup
• Far field approximation
• When nature is doing Fourier transform
• Far field diffraction pattern for common aperture functions

Optics

Main reference

• Lectures from Sander Konijnenberg (ASML)

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• Convex and concave lens
• Real and virtual image

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• Reflection
• Refraction and Snell's law

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• Free space propagation
• Single thin lens propagation
• Paraxial approximation
• Transfer matrix
• Imaging condition

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• Magnification factor
• Magnifier
• Microscope
• Telescope

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• Chief ray and marginal ray
• Aperture
• Depth of focus

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• The concept of phase
• Wave equation
• Complex notation
• Plane wave

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• Refraction and Snell's law from plane wave equation
• Total internal reflection and Evanescent field

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• Spherical wave equation

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• Huygens–Fresnel principle

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• Double slit (point source) far field pattern
• As addition of complex-valued fields approximated as plane waves
• As path difference over integer multiple of wavelength

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• Diffraction grating with slits of negligible width (point source)
• Diffraction grating with slits of non-negligible width
• Grating spectroscopy
• Chromatic resolving power (resolvance)
• Free spectral range
• Numerical aperture (NA)

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• Michelson interferometry
• Tilted beam interference
• Haidinger pattern

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• Decompose initial field into plane waves with angular spectrum
• Propagate plane waves with plane wave equation
• Reconstruct field after propagation using angular spectrum

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• Huygens' principle
• Propagation of field using Rayleigh-Sommerfeld (R-S) integral
• R-S integral under Fresnel approximation
• R-S integral under Fraunhofer approximation
• Short notation for quadratic phase factor
• When is Fresnel approximation valid
• When is Fraunhofer approximation valid
• Fresnel number