1. Scalar theory of diffraction
EE 231 Introduction to Optics
Scalar theory of diffraction
Lesson 11
Andrea Fratalocchi

2. 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.

3. The Scalar theory of diffraction

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

5. 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

6. 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:

7. 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:

8. 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

9. 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

10. 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.

11. 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

12. 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

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

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

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

16. 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"

17. 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:

18. 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?

19. 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:

20. 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

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

22. 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

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

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