1. Scalar theory of diffraction
EE 231 Introduction to Optics
Scalar theory of diffraction
Lesson 2
Andrea Fratalocchi

2. What is diffraction?

3. What is diffraction?
Slit experiment
Light focusing through a lens

4. What is diffraction?
Sommerfeld definition of diffraction:
Diffraction encompasses optical phenomena that cannot be interpreted by the propagation of rays following the laws of reflection and refraction in homogenous/in-homogenous media.
Slit experiment
Light focusing through a lens

5. Question: are these examples of diffraction?

6. Question: are these examples of diffraction? No
All these are effects of refraction of light rays. Our brain can only interpret light rays propagating along straight lines and creates false images when light is bent around curved trajectories

7. How to study diffraction?
By using Maxwell Equations, which are the fundamental set of equations to study all forms of light-matter interactions
The solution of the vectorial set of Maxwell equation is a quite challenging task in general, and in many cases cannot be done analytically.
Numerical computations can be carried out, but:
Require large simulation times and/or supercomputers
Give only numbers, making physical interpretations not simple
Depend on many degrees of freedoms, which makes it difficult to isolate governing physical quantities

8. How to study diffraction?
Powerful tools to address this problem are approximate theories, whose equations are easily solved even in the case of complex geometries. One of the most important approaches the scalar theory of diffraction. The advantages of this method are:
Simple and easy to use
Gives analytic functions, which make it easy to understand the physics of diffraction phenomena
Gives very good approximate results, which are in very good agreement with numerical computations

9. The Scalar theory of diffraction
The scalar theory of diffraction is based on some assumptions, which have to be clearly known in order to use this approach consistently and get reliable results
From Maxwell equations: 2
Wave equation

10. The Scalar theory of diffraction
The same equation is obtained for the magnetic field in a uniform medium:
If the electric field is linearly polarized and does not change polarization during the dynamics, we can model the evolution of the electromagnetic wave from a single scalar quantity U, following a scalar wave equation:
1st assumption
The field U(x,y,z;t) is often called disturbance field, or scalar field. The scalar field can equally represent the electric of magnetic field, as they both follow the same wave equation.

11. The Scalar theory of diffraction
In our study we will consider monochromatic waves:
2nd assumption
After substitution in the scalar wave equation, we obtain:
Helmholtz equation
This is the fundamental equation that describes scalar diffraction problems of light matter interactions.

12. The Scalar theory of diffraction
What is the intensity I(x,y,z) observable for the scalar field U?

13. The Scalar theory of diffraction
What is the intensity I(x,y,z) observable for the scalar field U?
In the scalar theory, the scalar field U can represents equally the electric or the magnetic field, and therefore all proportionality constants become unimportant. As we saw in the previous lesson, the intensity of an electromagnetic wave is proportional to the absolute square of the electric or magnetic field. In the scalar theory the intensity I(x,y,z) is therefore calculated as follows:

14. The Scalar theory of diffraction
The general diffraction problem:
Propagation problem
Propagation problem
Input field
Diffracting element
Observation screen
The problem is decomposed into 2 problems

15. The Scalar theory of diffraction
Propagation problem:
Given the spatial distribution of a scalar field V at a specific plane (x,y), propagate it to a specific z
Propagation problem
Input field
Diffracting element
This problem can be completely solved for arbitrary fields V

16. The Scalar theory of diffraction
Propagation problem:
Given the spatial distribution of a scalar field impinging on a diffractive object at z=0, calculate the emerging field at z=0
This problem is very hard and general solutions do not exist, however, approximate methods are available

17. The Scalar theory of diffraction
Homework 2.1
Propagation problem: interference of plane waves
At z=0, we have the following scalar field
Propagate the field at a generic z and calculate the correspondent Intensity
(Hint: use the plane wave solution derived in the previous lesson)

18. The Scalar theory of diffraction
Homework 2.2
Is this diffraction?
Motivate your answer

19. The Scalar theory of diffraction
References
M. Born and E. Wolf, Principle of Optics, 6th Edition, Chapter