Scalar Theory of Diffraction in Optics
The scalar theory of diffraction in optics as presented by Andrea Fratalocchi in this series of slides. Learn about the general solution to the propagation problem, the decomposition with Fourier Transform, and the propagation of single plane waves.
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EE 231 Introduction to Optics Scalar theory of diffraction Lesson 7 Andrea Fratalocchi www.primalight.org 3/18/2025 1 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 1
The Scalar theory of diffraction General solution to the propagation problem Sum up Propagate Decompose 3/18/2025 2 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 2
The Scalar theory of diffraction General solution to the propagation problem Plane wave Spatial frequencies Decompose with inverse Fourier Transform 3/18/2025 3 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 3
The Scalar theory of diffraction General solution to the propagation problem Decompose with Fourier Transform Propagate the single plane wave 3/18/2025 4 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 4
The Scalar theory of diffraction General solution to the propagation problem Propagate the single plane wave 3/18/2025 5 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 5
The Scalar theory of diffraction General solution to the propagation problem Propagate the single plane wave 3/18/2025 6 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 6
The Scalar theory of diffraction General solution to the propagation problem Propagate the single plane wave Sum up all the contributions 3/18/2025 7 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 7
The Scalar theory of diffraction Sum up Decompose Propagate 3/18/2025 8 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 8
The Scalar theory of diffraction Does not explicitly contains the input scalar field How can we express the output field as a function of the input? 3/18/2025 9 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 9
The Scalar theory of diffraction Does not explicitly contains the input scalar field How can we express the output field as a function of the input? Fourier transform of 3/18/2025 10 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 10
The Scalar theory of diffraction By substituting Into We finally obtain: 3/18/2025 11 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 11
The Scalar theory of diffraction We therefore have: 3/18/2025 12 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 12
The Scalar theory of diffraction We therefore have: According to the value of the spatial harmonics p and q, we can have 2 different scenario: 3/18/2025 13 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 13
The Scalar theory of diffraction According to the value of the spatial harmonics p and q, we can have 2 different scenario: Homogeneous plane wave with real propagation constant kz=2*pi*m 3/18/2025 14 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 14
The Scalar theory of diffraction According to the value of the spatial harmonics p and q, we can have 2 different scenario: Evanescent plane wave with imaginary propagation constant kz=2*pi*m 3/18/2025 15 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 15
The Scalar theory of diffraction Homework 1: consider the following scalar field 1) Evolve the field up to the distance z 2) Calculate the intensity. When evanescent waves start to appear? What are the physical consequences of this? 3/18/2025 16 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 16
The Scalar theory of diffraction Homework 2: According to visual experience, light sources (monochromatic or not), spread their content of energy during propagation more strongly if the region that emits light is more localized in space. Explain this phenomenon with the theory of diffraction 3/18/2025 17 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 17
The Scalar theory of diffraction References 1) M. Born and E. Wolf, Principle of Optics, Chapter 8. 3/18/2025 18 Andrea Fratalocchi (www.primalight.org) EE231 Introduction to Optics: Bessel beams slide 18
