ELG5106 Fourier Optics

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

2. Fourier Optics Diffraction 2

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

4. Plane Wave Expansion I Evanescent wave 4

5. Plane Wave Expansion II 5

6. Plane Wave Expansion III Spatial Frequency Response Impulse Response /Point Spread Function Linear ShiftInvariant System 6

7. Propagation as a filter u v unimodularphasefunction exponential decay 7

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

9. 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

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

11. 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

12. Paraxial Approximation II 12

13. 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

14. 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

15. 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.