Introduction to Diffraction

The plane wave picture of light is idealized in that it provides for light propagating to infinity in its tranverse dimensions, which of course is never realized in real life due to such effects as we will be discussing in this section. Diffraction describes the spreading of light once confined to a finite region, such as when light passes through an aperture. We will see that the specific way in which the light diffracts depends on the physical parameters of the confinement region, as well as the wavelength of the light being diffracted. The two methods of describing diffraction are Fresnel (and Huygens-Fresnel) and Fraunhofer diffraction.

Essentially, then, the problem is one of figuring out how to define light as a beam, or a path of light with varying transverse dimensions. This in effect means that we have to find a solution to the wave equation that accounts for these finite dimensions. Mathematically, there are two ways to accomplish this:

(1) Solve the Helmholtz or the paraxial wave equation directly to find beam-like solutions. This is a more useful method to describe how laser light may be confined.

(2) Solve the wave equation using an integral approach, which will then lead to the Fresnel and Fraunhofer diffraction integrals. This is a more useful technique for aperture problems.

With this, there are two different methods to describe diffraction: Fresnel and Fraunhofer diffraction. Fresnel diffraction describes near-field diffraction, while Fraunhofer diffraction describes the far-field (ie at many Rayleigh ranges away from the source). This is one of the principles used to make holography work. We will begin with Fresnel diffraction, and Huygen’s Principle.

With this, there are two different methods to describe diffraction: Fresnel and Fraunhofer diffraction. Fresnel diffraction describes near-field diffraction, while Fraunhofer diffraction describes the far-field (ie at many Rayleigh ranges away from the source). This is one of the principles used to make holography work. We will begin with Fresnel diffraction, and Huygen’s Principle.

First, though, we are here for an intuitive look at diffraction! There will be plenty of time for math later. To start with, if you haven’t already taken a look at the “Light as an Electromagnetic Wave” subsection of “Classical Optics”, I would recommend doing so to familiarize yourself with the angular spectrum representation of a field since that is a lot of what we will be doing now! The angular spectrum can be considered a general solution to the Helmholtz equation, which is what we said before we would intend to solve. It gives the distribution of the electric field in a specific plane, A:

\begin{equation} E_A(x,y,z) = \int\int_{-\infty}^{\infty}A(k_x,k_y)e^{-i(k_xx+k_yy+\beta{z})}dk_xdk_y \end{equation}

We will be using a property of linear response theory to describe the propagation of this wave in space, i.e. in the $\beta{z}$ direction, so it will be useful to recall that we defined our wave vector as:

\begin{equation} k = k_x\hat{x}+k_y\hat{y}+\beta{z} \end{equation}

which has the total magnitude:

\begin{equation} k^2 = k_x^2+k_y^2+\beta^2 = (\frac{2\pi}{\lambda})^2 \end{equation}

Now solving for $\beta$:

\begin{equation} \beta = \sqrt{k^2-k_x^2-k_y^2} \end{equation}

We will also recall that the meaning of $k_x$ is a spatial frequency in the $x$-direction, and same for $k_y$ in the y-direction. If the extent of these vectors in their respective directions is zero, then the spatial frequency in these directions is zero and the light is propagating purely in the $z$ direction. If, however, there is a nonzero spatial frequency in, for example, the $x$-direction, we can find the total angle of the wave with respect to the z-axis as:

\begin{equation} cos\theta_x = \frac{k_x}{k} \end{equation}

Also recall that the propagation of the plane along the z-axis was influenced by the spatial extent of the transverse spatial frequency components, i.e.

\begin{equation} k_x^2 + k_y^2 > (\frac{2\pi}{\lambda})^2 \end{equation}

leads to imaginary $\beta$ and there are then evanescent wave solutions confined near the plane A. In the opposite case, the propagation vector $\beta$ is real and will propagate away from the plane A to the right. With this, then, the meaning of the angular spectrum representation is that any field can be decomposed into plane waves of different directions propagating in the positive z-direction. This implies this would be a useful tool to mathematically describe diffraction!

Now, the angular spectrum i.e. the plane wave expansion is a useful tool because then the propagation of individual plane wave components to the right is trivial. If we set our origin to be z=0, the field at some plane $A(k_x,k_y,z)$ can be described as:

\begin{equation} A(k_x,k_y,z) = A(k_x,k_y; 0)e^{-i\beta{z}} \end{equation}

So now we can just define a transfer function as:

\begin{equation} H(k_x,k_;z) = e^{-i\beta{z}} \end{equation}

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