PHYS 352, Optics, Hope College, Spring 2013.
Regimes of Electromagnetic Circuits
 Kirchoff’s Laws
Microwaves
Optics
8
Particle
Important Trig Identities for this optics course
 Photonics
Three Regimes of Scalar Diffraction Theory
Figure from the 1998 Vienna PhD
thesis of Heinrich Kirchauer.
I recommend reading it!
Fourier Transform
2
3
SPECIAL FUNCTIONS
Bessel Functions
We will use Bessel functions for scalar diffraction theory and in Fourier optics.
Zeroth order Bessel function of the 1st kind:
J 0 (u ) 
1
2
2
e
iu cos(v )
dv
v 0
(Values are tabulated. Google Bessel Function Table. Bessel functions are also well known in all math
software. Play with it and get familiar.)
Useful integral identity that relates the 1st and 0th order Bessel functions:
u
J
0
(u ' )u ' du '  uJ 1 (u )
(u’ is a “dummy variable”)
u ' 0
And finally a very useful limit that will come in handy with diffraction patterns:
lim
u 0
J 1 (u ) 1

u
2
The sinc function
sinc(x) 
sin( x)
x
lim sinc(x)  1
x 0
The Gaussian
The Gaussian envelope: f ( x)  Ce
 ax2
(Equation 11.11)
The Fourier Transform of a Gaussian is a Gaussian. Never forget this. Also, because of the shape of a
laser resonator, the intensity of most laser beams has a Gaussian radial profile: I (r )  e 2 r
The Dirac Delta Function

 f ( x) ( x  a)dx  f (a)
The Fourier transform of a Dirac delta function is unity.

4
2
/ w2
.