Scalar theory of diffraction EE 231 Introduction to Optics Scalar theory of diffraction Lesson 11 Andrea Fratalocchi www.primalight.org

The Scalar theory of diffraction Homework 3: what about the phase term ? The total phase of the Gaussian beam reads as follows Phase term of a plane wave Gouy phase term The Gouy term represents a phase anomaly contribution to that of a plane wave solution. In most experiment this term is unobserved as we are looking at the intensity of the field. It plays a role for high order Gaussian beams.

The Scalar theory of diffraction

The Scalar theory of diffraction Exercise 1: calculate the power carried by a Gaussian beam

The Scalar theory of diffraction Exercise 1: calculate the power carried by a Gaussian beam From the definition of Power carried by an electromagnetic wave

The Scalar theory of diffraction Exercise 1: calculate the power carried by a Gaussian beam From the definition of Power carried by an electromagnetic wave We have:

The Scalar theory of diffraction Exercise 1: calculate the power carried by a Gaussian beam From the definition of Power carried by an electromagnetic wave We have:

The Scalar theory of diffraction Direct and inverse problem with Gaussian beams Direct Problem: Given the smallest spot-size , calculate all the parameters of the Gaussian beam

The Scalar theory of diffraction Direct and inverse problem with Gaussian beams Direct Problem: Given the smallest spot-size , calculate all the parameters of the Gaussian beam From the Rayleigh distance

The Scalar theory of diffraction Direct and inverse problem with Gaussian beams Inverse problem Given And At a distance z from an arbitrary origin, calculate And the position z' of the waist.

The Scalar theory of diffraction Direct and inverse problem with Gaussian beams Inverse problem Given And At a distance z from an arbitrary origin, calculate And the position z of the waist. In this case we need to solve the evolution equations for the spot-size and the curvature in terms of L and z

The Scalar theory of diffraction ABCD representation of Gaussian Beams We introduce the parameter q defined as follows: At z=0 Exercise 2: propagate the beam at generic z and calculate the equivalent q parameter

The Scalar theory of diffraction ABCD representation of Gaussian Beams Exactly like a translation matrix in ray optics!

The Scalar theory of diffraction ABCD representation of Gaussian Beams This is a general result Optical system

The Scalar theory of diffraction The interaction problem In the scalar theory of diffraction, the interaction problem is addressed by defining a ''transfer function"

The Scalar theory of diffraction The interaction problem The transfer function is defined by the ratio between the output scalar field and and input field impinging the diffractive object In the scalar theory of diffraction, the interaction problem is addressed by defining a ''transfer function"

The Scalar theory of diffraction The interaction problem Example: slit of width a centered at z=d If we assume that the material absorbs all impinging energy except the one passing through the slit, we have:

The Scalar theory of diffraction The interaction problem Example: slit of width a centered at z=d If we assume that the material absorbs all impinging energy except the one passing through the slit, we have: Question: what are the limitations of this approach?

The Scalar theory of diffraction The interaction problem Question: what are the limitations of this approach? It assumes that the response of the diffractive object does not depend on the input conditions but only on the geometry of the object. In general:

The Scalar theory of diffraction The interaction problem How big is this contribution? Let us evaluate it with an example This is a classical problem in diffraction theory that can be solved exactly with no approximation

The Scalar theory of diffraction The interaction problem The perfect conductor generates an oscillating field, that interferes with the impinging wave

The Scalar theory of diffraction The interaction problem This contribution is small and extends for few wavelengths only. This "second " order effect is neglected in the scalar theory of diffraction and in the transfer function approach at first order. The perfect conductor generates an oscillating field, that interferes with the impinging wave

The Scalar theory of diffraction The interaction problem Homework 1: calculate the transfer function of a thin lens

The Scalar theory of diffraction References A. Yariv, Optical electronics in modern communications, Chapter 2.