Integral Approach to Diffraction

In the previous sections, we found the formulations for how to describe a beam of finite transverse dimensions propagating in space by making the paraxial approximation, which it turned out is the same as the Fresnel approximation. We will now use this for the case of diffraction i.e. passage through some aperture. The standard treatment of diffraction theory can be found in many texts. We use some of the results of diffraction theory such as the angular spectrum formulation to get some key results, but some good reference texts on exactly why the angular spectrum is useful are given in the Rayleigh-Sommerfield diffraction formulas as well as in the Mandel+Wolf book. We will continue on with our approach for the time being to find the Fresnel diffraction formula.

Table of Contents:

  1. Angular Spectrum Approach
  2. Paraxial Diffraction

ANGULAR SPECTRUM APPROACH

In the case of diffraction through an aperture, we can take the aperture to be in the reference plane A at z = 0. We are given the field distribution in the aperture, and wish to calculate the field on the observation screen at z. If this sounds familiar, that’s great, because we have in fact been solving this type of problem in the last few sections! If not, then maybe look at the last few sections, but I will just tell you that the approach this then sets us up for is the angular spectrum approach to diffraction. The field at some propagation distance z is obtained by multiplying the angular spectrum by the transfer function of free space $e^{-i\beta{z}}$. The field in real space can then be obtained via an inverse 2D Fourier transform. An equivalent method would be to convolve the field E(x,y;0) with the impulse response function of free space.

Mathematically, the angular spectrum in the aperture is given as:

\begin{equation} A(k_x,k_y;0) = \frac{1}{4\pi^2}\iint_{-\infty}^{\infty}E(x’,y’;0)e^{i(k_xx’+k_yy’)}dx’dy’ \end{equation}

so then the plane at a distance z is:

\begin{equation} A(k_x,k_y;z) = H(k_x,k_y;z)A(k_x,k_y;0) \end{equation}

or

\begin{equation} E(x,y,z) = E(x’,y’;0)\circledast h(x,y;z) \end{equation}

where “$\circledast$” implies the convolution.

PARAXIAL DIFFRACTION

We saw before that in the Fresnel or paraxial approximation, we could rewrite the propagation k-vector $\beta$ such that we have the impulse response (Fourier transform of transfer function of free space):

\begin{equation} h(x,y;z) \approx -\frac{e^{-ikz}}{i\lambda{z}}e^{-ik(x^2+y^2)/2z} \end{equation}

Then we can write out the convolution integral explicitly as:

\begin{equation} E(x,y;z) = \iint^{\infty}_{-\infty}E(x’,y’;0)e^{-ik[(x-x’)^2+(y-y’)^2]/2z}dx’dy’ \end{equation}

This is a huge result! This is the Fresnel diffraction formula, which governs light diffraction in the near field (i.e. within a few Rayleigh ranges of the aperture). Let’s review how we got here:

  1. Recognized the need for a method to describe a wave with transverse dimensions i.e. a beam.
  2. Helmholtz equation for time independent propagation is generally solved by the angular spectrum. Since the angular spectrum basically breaks down a field in a plane into its constituent plane wave components propagating in different directions along the axis, this implied it would be good to figure out how a wave with nonzero finite dimensions is affected by transverse confinement.
  3. We found the transfer function of free space, which was just an exponential term for propagation along z.
  4. We applied the Fresnel, and then the paraxial approximation that the values for the rate of range of the envelope in the x and y directions is much slower than in the z-direction, so we could rewrite some of the propagation constants into a much simpler form.
  5. Putting it all together, we were able to take the field in an aperture and propagate it to the near field using the results from above! Our main result is that the paraxial diffracted field is a convolution of the aperture field with a quadratic phase advance in x- and y-directions.
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