The Fraunhofer Plane



The Basics of Diffraction and the Fraunhofer Plane:


     Diffraction occurs when light deviates from a straight line path and enters a region that would otherwise be shadowed.  This deviation from general optics behavior occurs when waves pass through small openings, around obstacles, or past sharp edges.  These patterns can only be described by using the wave theory of light which predicts these "smeared" areas of constructive and destructive interference.  There are, however, two basic types of diffraction situations, both due to different conditions.  These two types of diffraction are called Fresnel and Fraunhofer diffraction, the latter being what is discussed on this web page. 

    Fraunhofer diffraction occurs when both incident and diffracted waves are plane (see description of plane waves here).   To create such a situation, one must make the distance from the light source to the diffracting obstacle to the observation point large enough to neglect the curvature of incident and diffracted light.  In other words, referring back to Huygen's Principle, when the light reaches the diffracting aperture, the spherical wavefront should be large enough that it is virtually a planar wave front (basically a flat, vertical line) over it.   The rays must then be parallel, or close to parallel, as they reach the diffracting object.  The point at which the diffraction pattern is observed, in the case of the experiment detailed below the focal length of a converging lens, becomes the Fraunhofer plane. 

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The Fourier Transform and Diffraction in the Fraunhofer Plane:


   The Fourier Transform, as it turns out, proves to be a  powerful tool when it comes to describing and analyzing diffraction patterns in the Fraunhofer Plane.   Consider the setup pictured below:

fouriersetup.gif (6747 bytes)

    The diffracting obstacle in the picture above resides in the xy plane and the lens focuses the diffracted rays to the XY, or Fraunhofer, plane.  From the properties of a converging lens, all light leaving the aperture and passing through the lens will be brought to a common focus in the XY plane at the focal length, f, of the lens.  In the diagram above, light from the point Q(x,y) will be focused at point P(X,Y).  The X and Y coordinates at P can be described using direction cosines and are denoted as  and Now consider again that ray originating at Q and travelling to parallel to O and arriving at P.   The difference in length between OO and QP can be determined from the diagram below:

gammar.gif (3229 bytes)

    The path difference is denoted by the symbol wpe9C.jpg (748 bytes) and represents the dot product of the position vector from the point O to Q, R, with the unit normal vector to O, n.  R exists in the xy plane and is written The unit normal, n, is written in terms of direction cosines From the picture above, and as mentioned before,  wpe9C.jpg (748 bytes) is the dot product of R with n so With a little intuition, one recognizes a the relationship of our situation to the Fresnel-Kirchoff formula that describes how a diffraction pattern can be obtained by integrating the phase of incident light over the aperture.  The Fresnel-Kirchoff formula is stated as follows:

integral.gif (1083 bytes)

We can now extend this equation to the diffraction pattern in our XY plane using the path difference we estimated above.  The following modifications are made to the Fresnel-Kirchoff equation:

The phase difference of the light would be due to the distance wpe9C.jpg (748 bytes) so we substitute it in the formula for the corresponding variable r.  Also, the dA is now over the   xy plane where our aperture exists, so dA = dxdy.  This integral is in terms of the spatial frequencies denoted by and   which have units in terms of reciprocal length, not time.  This function U(X,Y) determines what the amplitude of the diffraction pattern will look like in the XY plane.  The model, however, assumes a uniform, perfect, diffracting aperture.  This being an idealistic situation,  a function that can take into account these imperfections must be introduced.  This compensatory function, denoted g(x,y), is called the aperture function and describes the diffracting object, usually using Fourier Series sines and cosines.  Literally, g(x,y) represents the amplitude of the scattered light originating from the area element dxdy (remember Huygen's principle says one can consider each point on the aperture to be a source for secondary wavelets whose coherent superposition will give us the diffraction pattern in the XY plane).  To take into account the aperture function, we rewrite the Fresnel-Kirchoff formula again in terms of a Fourier Transform integral:

Essentially, this reveals that the interference pattern observed at the observation point P is a Fourier resolution of the aperture function g(x,y).  Thus, the diffraction maxima are composed of Fourier components.  If some of the components are removed from the XY/Fraunhofer plane, interesting effects occur resulting from this spatial filtering. 

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Spatial Filtering in the Fraunhofer Plane:


    An interesting phenomenon occurs as one views the image plane of the Fraunhofer setup.  Consider the diagram below:

fouriertrans.gif (8620 bytes)

    The xy plane is where the aperture resides (as in the picture in the theory section), so our diffraction pattern determined by the amplitude function g(x,y) and Fresnel-Kirchoff formula U(x,y) appears in the uv plane (same as XY plane described in theory section).  Thus, to see the image that appears in the x'y' plane in the diagram above, which should be of the diffracting aperture itself, one needs to integrate the phase of light over the Fraunhofer Plane (as was done in the xy plane) to obtain the diffracting object pattern.  This results in an inverse Fourier Transform integral and is clearly similar to the steps taken to determine the interference pattern before.  Therefore, we designate the amplitude function as g(x',y') since the image will appear in the x'y' plane and should be the exact same as the aperture function g(x,y) (assuming ideal conditions).  Ideal conditions in lab, however, are virtually non existent!  Some information, or spatial frequencies when thinking in terms of Fourier Transforms,  will always be lost in the process of diffracting to recombining the light in the image plane.  This again is due to the wave nature of light and the definition of diffraction.  One must introduce a function, call it T(u,v), the Transfer function,  that accounts for this loss of information whether it be from the scattering of light of lens abbhorrations.  Finally, one is able to write out this inverse Fourier Transform integral describing the information that appears to the observer in the image plane.  The integral is written as follows:

    The image amplitude function g'(x',y') should really be similar to the aperture function g(x,y) while U(u,v) is the diffraction pattern amplitude in the uv plane. 

    Spatial filtering may be achieved by removing information in the Fraunhofer Plane (uv plane).  To do this, spatial frequencies must be removed so that what the observer sees in the image plane isn't entirely the diffracting object, but selected information about it.  This can be done by placing objects in the Fraunhofer plane that prevent certain spatial frequencies from passing.  In the experiment performed and described below, we do so to the diffraction patterns of screen wire, single slit, and circular diffracting apertures.   

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Experiments With the Fraunhofer Plane:


Our Setup (Thank you Seth Carpenter for the image):

ppt-setup.gif (8591 bytes)

 

     The diffraction amplitude integrals are fairly easily modeled using Mathematica.  The first example, and simplest, is the single slit example.   If we set up the situation mathematically as follows

singleslit.jpg (7219 bytes)

our Fourier Transform Integral becomes Image104.gif (1245 bytes) (assuming constant aperture function of 1) which when integrated yields Image105.gif (1659 bytes).  This solution can be modeled with Mathematica as seen below:

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This is the code that we used to model the single slit.  The bottom picture represents a 3-D realization of the intensity of the diffraction pattern using Mathematica.  There is the one large central maxima and then the less intense ones around it. We determined our slit width and length experimentally as with the other constants.  The output looks as expected for single slit diffraction. Here is a picture of the actual diffraction pattern in the Fraunhofer plane from the experiment using a HeNe laser as the light source:

slit-diff.jpg (2389 bytes)

   Now, let's do a reverse Fourier Transform.and we should get back the an image of the slit itself!

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We do get back the image of the slit!  It is just a constant function.  But what happens when we vary our aperture function g(x,y) to something other than a constant?

    Let's make our single slit, initially a flat planar square, now a cosine function.  We should guess that this change will affect the maxima around the central one.  Since the edges of the slit are no longer sharp, we remove some of the edge information components of the Fourier transform.  Let's see what happens:

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As  predicted, the outer maxima are significantly reduced in relation to the central peak.  Unfortunately, we get an non convergence error when trying to do the Inverse Fourier transform to see what our diffracting aperture looks like in the image plane with information removed.  However, this was done experimentally!   In the first case of spatial filtering, we removed all but the central maxima of the diffraction pattern in the Fraunhofer plane.  Using a video camera, we captured a picture of the diffracting aperture with information removed in the image plane.  The picture is seen below:

diff1.gif (2963 bytes)             diff3.gif (2893 bytes)

    The picture on the left represents an image of the slit with no information removed.  Notice how its edges are sharp and well defined!   The picture on the right is the image with the outer maxima removed.  Remark how it is very blurred and not as defined as the slit with no spatial frequencies removed.   This is analogous to the removing the edge information (higher components of a Fourier series). 

wpe133.jpg (5516 bytes)         wpe134.jpg (5352 bytes)

    The picture on the left represents the Fourier series of a square wave, but only the first 2 partial sums of the series.  Notice how blurry and indistinct it really is.  The picture on the right represents the square wave with many components and a lot of edge information. There clearly is a relationship to the Fourier series and to how spatial filtering works!

For our next attempt at spatial filtering, we removed the central maxima in the single slit diffraction pattern and left all of the edge information.  This is what we saw in the image plane:

diff2.gif (2853 bytes)

Notice how the slit still remains somewhat sharp, but all the edge information is gone from the center.  Lets look at the analogy to the Fourier Series again:

wpe135.jpg (7161 bytes)

Above represents a picture of the same Fourier series of a square wave with the first several components removed and all the edge information left. See how the top portion of the square wave disappears, but the outside edges are left.  This situation is exactly what we observe in the image plane when the central peak of the diffraction pattern is removed!

    Lets see what happens when a pinhole is the diffracting aperture!  We set the situation up mathematically as follows:

circular.gif (4734 bytes)

When the mathematics are set up as shown, the following Fourier Transform Integral describing the diffraction pattern amplitude is obtained Image107.gif (1352 bytes).  The solution to this is a 1st order Bessel function.  The solution looks something like this when evaluated in mathmatica:

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Experimentally, we observed the following diffraction pattern:

    pinhole.jpg (2595 bytes)

    The mathematical model coincides with the image that we got experimentally.   Notice the bright central maxima surrounded by the outer rings smaller in intensity.  We can also observe what the aperture looks like in the image plane for this experiment as well.  We took the following picture with our video camera in the image plane for the pinhole aperture:

diff7.gif (1352 bytes)

Again, it resembles a pinhole as our theory would suggest.  

    Finally, Seth Carpenter and I performed this experiment using a piece of chicken wire as our diffracting aperture.  Without a spatial filter we obtained the following picture in the image plane:

diff4.gif (13665 bytes)

This very much resembles the piece of wire that we used as the diffracting object.  Next, we proceeded to remove information from the diffraction pattern in the Fraunhofer plane.  First, we removed all of the horizontal information leaving several vertical striped.   The following picture was obtained in the image plane:

diff5.gif (6722 bytes)          screen-diff2.gif (5122 bytes)

The picture on the left represents what we saw when we removed the information in the Fraunhofer plane using the method in the picture on the right.  We merely placed a piece of black paper across the horizontal portion of the diffraction pattern in the Fraunhofer plane.  If you look closely at the vertical lines, however, you might notice that some of the horizontal information passes through as there exist horizontal marks in the vertical patterns. 

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Other Applications of Diffraction and the Fraunhofer Plane:


    The ability of optical systems to distinguish  between closely spaced objects is limited because of the wave nature of light.  Consider the example of a single slit of width dx.  If the slit becomes illuminated by two difference sources, the light reaching the slit will be incoherent.  If the slit is wide enough, two distinct images will appear to the on the observation plane (see picture below).   Due to the effects of diffraction, however, each image will be comprised of maxima and minima so one actually sees a sum of these two diffraction patterns.  Notice that if the two sources (S1 & S2) are far enough apart to keep their central maxima from overlapping, then their images will be distinguishable to the viewer at the observation plane. If the sources are moved close together, their maxima overlap and the two images become indistinguishable from one another. This parallels the theory behind the microscope. One can only resolve whatever one is looking at so far.  For instance, if we had a small cell, we could resolve the cell walls, but not the DNA inside it.

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References:

  1. Fowles, Grant R.  Introduction to Modern Optics.   Dover Publications: New York.  1989.
  2. Serway, Raymond A.  Physics For Scientists and Engineers with Modern Physics.  Saunders College Publishing:   Philadelphia.  1996. 
  3. Tolansky, S.  An Introduction to Interferometry. William Clowes and Sons Ltd.:  London.  1966.

Table of Contents:

  1. Abstract
  2. History of Wave Theory and Optics
  3. Michelson Interferometer
  4. Travelling Mirror Michelson Interferometer
  5. Fabry-Perot Interferometer

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