1. Scalar theory of diffraction
EE 231 Introduction to Optics
Scalar theory of diffraction
Lesson 12
Andrea Fratalocchi

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

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

4. The Scalar theory of diffraction
The interaction problem
Homework 1: calculate the transfer function of a thin lens
From the definition of the transfer function
Input wave:
Output wave:

5. The Scalar theory of diffraction
The interaction problem
Homework 1: calculate the transfer function of a thin lens
From the definition of the transfer function
In paraxial approximation:

6. The Scalar theory of diffraction
The interaction problem
The transfer function of a thin lens depends only on one degree of freedom, the focal length f

7. The Scalar theory of diffraction
The interaction problem
Application of lenses
Far field observation
According to the Far-Field approximation, the Far-Field is observed when

8. The Scalar theory of diffraction
The interaction problem
Application of lenses
Far field observation
At a wavelength of 500 nm, and for an object of area of 1mm^2, we have:

9. The Scalar theory of diffraction
The interaction problem
Is there a better way to visualize the Far-Field?
Application of lenses
Far field observation
At a wavelength of 500 nm, and for an object of area of 1mm^2, we have:
This distance is quite large, especially in a lab where the characteristic lengths of optical tables are 2-3 meters

10. The Scalar theory of diffraction
The interaction problem
Application of lenses
Exercise 1: given the input field Vi impinging on a thin lens of focal f
Far field observation
Calculate the output intensity on the focal plane at z=f

11. The Scalar theory of diffraction
Exercise 1: given the input field Vi impinging on a thin lens of focal f
The interaction problem
Application of lenses
Far field observation
Calculate the output intensity on the focal plane at z=f

12. The Scalar theory of diffraction
Exercise 1: given the input field Vi impinging on a thin lens of focal f
The interaction problem
Application of lenses
Far field observation
Calculate the output intensity on the focal plane at z=f
This simplifies into:

13. The Scalar theory of diffraction
Exercise 1: given the input field Vi impinging on a thin lens of focal f
The interaction problem
Application of lenses
Far field observation
Calculate the output intensity on the focal plane at z=f
Fourier Transform of the input field!

14. The Scalar theory of diffraction
Exercise 1: given the input field Vi impinging on a thin lens of focal f
The interaction problem
Application of lenses
Far field observation
Calculate the output intensity on the focal plane at z=f
The lens visualizes a scaled version of the Fourier transform of the input field

15. The Scalar theory of diffraction
The interaction problem
Application of lenses
Problem: the lens visualizes the Fourier transform in Intensity, but the amplitude of the output field has a spherical curvature. This implies that the Fourier transform is imaged on a sphere. This is not the most desirable scenario when more lenses are employed, as the spherical term propagates as well
How to address this issue?
Far field observation

16. The Scalar theory of diffraction
The interaction problem
Application of lenses
Exercise 2: calculate the output field Vo
Use this configuration

17. The Scalar theory of diffraction
The interaction problem
Application of lenses
Exercise 2: calculate the output field Vo
Use this configuration
Simple proportionality!
In optics, the Fourier transform of a field is an experimental observable

18. The Scalar theory of diffraction
This is the most basic setup for the elaboration of optical signal in free space
The interaction problem
Application of lenses
2. 4-f optical systems
Exercise: calculate the output field of the system
The diffractive object acts as a filter with transfer function
With this setup, we can design the transfer function in real space!

19. The Scalar theory of diffraction
The interaction problem
Application of lenses
3. Collimation and focusing of Gaussian beams
Exercise 3: calculate the emerging field from the lens. Is the emerging field still Gaussian?

20. The Scalar theory of diffraction
The interaction problem
Exercise 3: calculate the emerging field from the lens. Is the emerging field still Gaussian?
Application of lenses
3. Collimation and focusing of Gaussian beams

21. The Scalar theory of diffraction
The interaction problem
Exercise 3: calculate the emerging field from the lens. Is the emerging field still Gaussian?
Application of lenses
3. Collimation and focusing of Gaussian beams
The output beam is still Gaussian, but with a different curvature

22. The Scalar theory of diffraction
The interaction problem
Exercise 4: calculate the ABCD matrix of the lens
Application of lenses
3. Collimation and focusing of Gaussian beams

23. The Scalar theory of diffraction
The interaction problem
Exercise 4: calculate the ABCD matrix of the lens
Application of lenses
3. Collimation and focusing of Gaussian beams

24. The Scalar theory of diffraction
The interaction problem
The Gaussian beam emerging from the lens is collimated, i.e., has in the lens plane the waist plane of minimum spot-size
Application of lenses
3a. Collimation of Gaussian beams
What are the conditions for this type of interaction?

25. The Scalar theory of diffraction
The Gaussian beam emerging from the lens is collimated, i.e., has in the lens plane the waist plane of minimum spot-size
The interaction problem
Application of lenses
3a. Collimation of Gaussian beams
What are the conditions for this type of interaction?

26. The Scalar theory of diffraction
The emerging Gaussian beam is focused at a point located at z=z' measured from the lens
The interaction problem
Application of lenses
3b. Focusing of Gaussian beams
How to find the new plane of minimum waist?

27. The Scalar theory of diffraction
The emerging Gaussian beam is focused at a point located at z=z' measured from the lens
The interaction problem
Application of lenses
3b. Focusing of Gaussian beams
How to find the new plane of minimum waist?
This is an inverse problem of Gaussian beams

28. The Scalar theory of diffraction
The emerging Gaussian beam is focused at a point located at z=z' measured from the lens
The interaction problem
Application of lenses
3b. Focusing of Gaussian beams
How to find the new plane of minimum waist?
At z=0, we have
By solving an inverse problem, we can calculate the position of the new waist of the emerging beam

29. The Scalar theory of diffraction
The emerging Gaussian beam is focused at a point located at z=z' measured from the lens
The interaction problem
Application of lenses
3b. Focusing of Gaussian beams
How to find the new plane of minimum waist?
At z=0, we have
By solving an inverse problem, we can calculate the position of the new waist of the emerging beam

30. The Scalar theory of diffraction
The interaction problem
Homework 1: the operations of focusing and collimation are often used together in the following setup
Lens A focus, while lend B collimates the beam. This is a Beam expander. Write the input output relationship of the system and the magnification factor

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