Trevor Hall tjhall@uottawa.ca ELG5106 Fourier Optics Trevor Hall tjhall@uottawa.ca

Fourier Optics Diffraction 2

Propagation between Planes in Free Space x1 x2 y1 y2 x3 x3=0 x3=z k 3

Plane Wave Expansion I Evanescent wave 4

Plane Wave Expansion II 5

Plane Wave Expansion III Spatial Frequency Response Impulse Response /Point Spread Function Linear Shift Invariant System 6

Propagation as a filter u v unimodular phase function exponential decay 7

Why is the angular spectrum of plane waves expansion rarely used? 8

Oscillatory Integrals We are left with the consideration of integrals of the form: If then the integrand is highly oscillatory and If then there is a contribution from the integrand in the neighbourhood of the stationary point p* 9

Stationary Phase Condition The stationary phase condition corresponds to a ray from source point to observation point ( recall shift invariance) 10

Paraxial Approximation I In a paraxial system rays are inclined at small angles to the optical axis. One may then make the paraxial approximation: 11

Paraxial Approximation II 12

Fresnel Diffraction Up to a multiplicative quadratic phase factor (that is often neglected), the field at the observation plane is given by the Fourier transform of the field at the source plane multiplied by a quadratic phase factor. 13

Fraunhoffer Diffraction If the source filed u has compact support (is zero outside some bounded aperture) and z is sufficiently large the variation of the quadratic phase factor over the support of u becomes negligible. The leading phase factor is also often neglected either because the region of interest in the observation plane subtends a sufficiently small angle with respect to the origin at the source plane or because it is the intensity only that is observed. The diffracted field distribution is then given by a Fourier transform of the field distribution in the source plane. 14

Notes The oscillatory integral representation of the impulse response of this optical system can be evaluated asymptotically without recourse to the paraxial approximation using the method of stationary phase. The magnitude but not the phase of the leading multiplicative phase factors of the Fresnel and Faunhoffer diffraction integrals may be evaluated by appealing to energy conservation – the integral over the source and observation planes of the field intensity must be equal. The choice of outgoing plane waves in the plane wave spectrum ensures that all three diffraction integrals (plane wave expansion, Fresnel & Fraunhoffer formulae satisfy the Sommerfeld radiation condition at infinity.