1. Scalar theory of diffraction
EE 231 Introduction to Optics
Scalar theory of diffraction
Lesson 5
Andrea Fratalocchi

2. The Scalar theory of diffraction
Homework 1: How many interference fringes N are found in a disc of radius a?
Do the calculation in paraxial and non paraxial approximation. Compare N in the case of R=1m, wavelength 500nm, disc radius a=20mm for paraxial and non paraxial calculations. What is the difference between the two cases?

3. The Scalar theory of diffraction
Homework 1: How many interference fringes N are found in a disc of radius a?
Two maxima are separated by an angular distance
The number of maxima N in a disc of radius a is therefore:

4. The Scalar theory of diffraction
Homework 1: How many interference fringes N are found in a disc of radius a?
Two maxima are separated by an angular distance
The number of maxima N in a disc of radius a is therefore:
For R=1m, wavelength 500nm and a=20mm, the number of fringes is already N=400!

5. The Scalar theory of diffraction
Homework 1: How many interference fringes N are found in a disc of radius a?
Added a phase shift to put in r=0 the point of maximum intensity
Exact calculation:
The number N of fringes contained in a disc of radius a is then:

6. The Scalar theory of diffraction
Homework 1: How many interference fringes N are found in a disc of radius a?
Added a phase shift to put in r=0 the point of maximum intensity
Exact calculation:
The number N of fringes contained in a disc of radius a is then:
With the same numbers as before, we get N=399.96=400. Paraxial approximation gives exactly the same

7. The Scalar theory of diffraction
Propagation of continuum ensembles of waves
Any combination of plane waves represents an electromagnetic field solution of Maxwell equations and, at such, can generate fields of interest
An important case is represented on the left: the propagation of a superposition of plane waves that possess wave vectors lying on a cone of semiaperture
At z=0, each plane wave is:

8. The Scalar theory of diffraction
Propagation of continuum ensembles of waves
Any combination of plane waves represents an electromagnetic field solution of Maxwell equations and, at such, can generate fields of interest
At z=0, each plane wave is:
In polar coordinates:

9. The Scalar theory of diffraction
Propagation of continuum ensembles of waves
Any combination of plane waves represents an electromagnetic field solution of Maxwell equations and, at such, can generate fields of interest
In polar coordinates:
At z=0, each plane wave is:

10. The Scalar theory of diffraction
Propagation of continuum ensembles of waves
Any combination of plane waves represents an electromagnetic field solution of Maxwell equations and, at such, can generate field of interest
Summing up all the contributions:
We begin by considering A constant

11. The Scalar theory of diffraction
Propagation of continuum ensembles of waves
Any combination of plane waves represents an electromagnetic field solution of Maxwell equations and, at such, can generate fields of interest
Recalling some results from basic calculus and special functions
Bessel function of order n

12. The Scalar theory of diffraction
Propagation of continuum ensembles of waves
Bessel function of order n
This substitution makes the two integral the same

13. The Scalar theory of diffraction
Propagation of continuum ensembles of waves
0-th order Bessel beam

14. The Scalar theory of diffraction
Propagation of continuum ensembles of waves

15. The Scalar theory of diffraction
Exercise: propagate the following Bessel beam at arbitrary z, do you find some interesting property?

16. The Scalar theory of diffraction
Exercise: propagate the following Bessel beam at arbitrary z, do you find some interesting property?
1. Decompose into known waves

17. The Scalar theory of diffraction
Exercise: propagate the following Bessel beam at arbitrary z, do you find some interesting property?
1. Decompose into known waves

18. The Scalar theory of diffraction
Exercise: propagate the following Bessel beam at arbitrary z, do you find some interesting property?
1. Decompose into known waves
2. Propagate each wave

19. The Scalar theory of diffraction
Exercise: propagate the following Bessel beam at arbitrary z, do you find some interesting property?
1. Decompose into known waves
2. Propagate each wave

20. The Scalar theory of diffraction
Exercise: propagate the following Bessel beam at arbitrary z, do you find some interesting property?
1. Decompose into known waves
2. Propagate each wave
3. Sum them up

21. The Scalar theory of diffraction
Exercise: propagate the following Bessel beam at arbitrary z
do you find some interesting property?

22. The Scalar theory of diffraction
Exercise: propagate the following Bessel beam at arbitrary z, do you find some interesting property?
do you find some interesting property?
Bessel beams are nondiffracting: intensity is constant along propagation and does not change!

23. The Scalar theory of diffraction
Homework 1: how to experimentally generate a Bessel beam?
Homework 2: how to superimpose plane waves and generate high order Bessel beams Jn(kr)? What are their interesting properties?
Homework 3: can you think of some possible applications of Bessel beams?

24. The Scalar theory of diffraction: Bessel Beams
References
D. McGloin, K. Dholakia, Bessel beams: diffraction in a new light, Contemporary Physics 46 (2005)