Site hosted by Angelfire.com: Build your free website today!

Fourier Optics

Home Up Next

Wave Equations And Fourier Optics

These notes have been converted to electronic format and additional text has been added for clarification.

 

Note

Paraxial Diffraction Approximation

Large and small z

 

 

 

Lecture 7 - Tuesday 26 May 2009 (Scott Parkins)

Electromagnetic waves are usually described in terms of transverse electric and magnetic vector fields but a scalar field description can also be useful. For this, we introduce the Laplacian Operator to describe a scalar wave equation for propagation in free space;  

        ...(1)

The Laplacian Operator (applied in this example to the electromagnetic field) is defined as

        ...(2)

Consequently, equation (1) can be expressed as

        ...(3)

Equation (3) shows a relationship between spatial characteristics and time characteristics for the wave function phi. We will use the familiar relationship between wavelength and frequency for a coherent, monochromatic wave

        ...(4)

Since this wave is coherent, it can be represented as a complex time exponential waveform beginning from a time origin at t=0

        ...(5)

This leads us to the "Helmholtz Equation" obtained by substituting equation (5) into equation (3)

        ...(6)

Note

The second time differential transforms e-j· 2 · pi · f · t   -»   -j· 2· pi · f · e-j·2·pi·f· t   -»  -(2· pi · f )2· e-j·2·pi·f· t  and from equation (4) we obtain -(2· pi · f/c )2· e-j·2·pi·f· t  -»  -(2 · pi /lambda )2· e-j·2·pi·f· t   -»  -k2· e-j·2·pi·f· t  were k = 2 · pi /lambda

Now consider propagation in one dimension (1-D) parallel to the z-axis. We describe this arrangement as paraxial

        ...(7)

We now consider spatial behavior - the time dependency is already described by equation (5). Note that the subscript z for fz(x, y) is just that - i.e. it does not represent a partial derivative.

We expect that   will vary slowly with z. If we consider a plane wave parallel to the z-axis, then  will be independent of z (i.e. parallel waves remain as parallel waves).

We can now substitute equation (7) into the Helmholtz Equation (6)

        ...(8)

The RHS term appears also on the LHS and cancels. Also we will neglect the 2nd derivative of the slow variation term fz with respect to z but will retain the 1st derivative (expected to be much larger) term. This results in a simpler expression for equation (8)

        ...(9)

We now need to solve this differential equation. To facilitate this let's consider a 2-D Fourier Transform (FT) and its inverse FT defined as

        ...(10)

Note:

This transform is somewhat abstract compared to the familiar 1 D correspondence between time and frequency. In this case, the variable x and y represent distances and the Fourier Transform variables u and v have no obvious physical meaning. For now, I will just consider these as transformation pairs.

We also note the transform pair for the 2nd derivative

        ...(11)

Taking the 2-D FT lets us rewrite equation (9) in simpler terms

        ...(12)

This represents a 1st order differential equation with separable variables

        ...(13)

The initial distribution  and its transform pair now take a Gaussian form 

        ...(14)

We can interpret this relationship as a convolution

        ...(15)

(see 8.14 on paraxial diffraction integral)

To summarize we specify fz (x,y) @ z=0 and solve for other z.

 

Example - Point Light Source From Pinhole in Opaque Material   

       

Interestingly, we consider the point source to be equivalent to 2 impulse functions as we travel in x and y directions.

        ...(16)

So 

        ...(17)

We can consider an exact solution to a point source,

        ...(18)

This square root has a binomial expansion for large z (paraxial approximation)

        ...(19)

       

We note that f is only weakly dependent on z

            ...(20)

Therefore 

        ...(21)

       

Now suppose again that z is large, i.e.

        ...(22)

We now encounter the Fraunhofer Approximation

        ...(23)

This has an intensity distribution given by

        ...(24)

 

Paraxial Diffraction Approximation

         

Fraunhofer Approximation

        ...(25)

   The Pz(x0 ,y0 ) term inside the double integral from equation (21) tends to 1 i.e. . Equation (21) now simplifies to

     ...(26)

Equation (26) now represents a 2-D Fourier Transform of f0(x0 , y0). 

       

       

       

       

       

       

       

 

       

       

       

 

   

       

       

       

       

 

       

       

 

       

       

       

       

       

       

       

       

       

       

       

       

       

 

Summary

 

 

Home Up Next

© Ian Scott 2009