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Diffraction formalismAdditional recommended knowledge
Quantitative description and analysisBecause diffraction is the result of addition of all waves (of given wavelength) along all unobstructed paths, then usual procedure is to consider contribution of infinitestimally small neighborhood around certain path (this contribution is usually called "wavelet") and then integrate over all paths (=add all wavelets) from the source to the detector (or given point on a screen). Thus in order to determine the pattern produced by diffraction the phase and the amplitude of each of the wavelets is calculated. That is, at each point in space, we must determine the distance to each of the simple sources on the incoming wavefront. If the distance to each of the simple sources differs by an integer number of wavelengths, all the wavelets will be in phase, resulting in constructive interference. If the distance to each source is an integer plus one half of a wavelength, there will be complete destructive interference. Usually, it is sufficient to determine these minima and maxima to explain the effects we see in nature. The simplest descriptions of diffraction are those in which the situation can be reduced to a 2 dimensional problem. For water waves, this is already the case, water waves propagate only on the surface of the water. For light, we can often neglect one dimension if the diffracting object extends in that direction over a distance far greater than the wavelength. In the case of light shining through small circular holes we will have to take into account the full three dimensional nature of the problem. General diffractionSeveral qualitative observations can be made of diffraction in general:
ApproximationsThe problem of calculating what a diffracted wave looks like, is the problem of determining the phase of each of the simple sources on the incoming wave front. It is mathematically easier to consider the case of far-field or Fraunhofer diffraction, where the point of observation is far from that of the diffracting obstruction, and as a result, involves less complex mathematics than the more general case of near-field or Fresnel diffraction. To make this statement more quantitative lets consider a diffracting object at the origin that has a size If we now consider the situation where This is the Fresnel approximation. To further simplify things, if the diffracting object is much smaller than the distance Depending on the size of the diffraction object, the distance to the object and the wavelength of the wave, the Fresnel approximation, the Fraunhofer approximation or neither approximation may be valid. As the distance between the measured point of diffraction and the obstruction point increases, the diffraction patterns or results predicted converge towards those of Fraunhofer diffraction, which is more often observed in nature due to the extremely small wavelength of visible light. Diffraction from an array of narrow slitsA simple quantitative description
Multiple-slit arrangements can be mathematically considered as multiple simple wave sources, if the slits are narrow enough. For light, a slit is an opening that is infinitely extended in one dimension, which has the effect of reducing a wave problem in 3D-space to a simpler problem in 2D-space.
The simplest case is that of two narrow slits, spaced a distance Maxima in the intensity occur if this path length difference is an integer number of wavelengths.
The corresponding minima are at path differences of an integer number plus one half of the wavelength:
For an array of slits, positions of the minima and maxima are not changed, the fringes visible on a screen however do become sharper as can be seen in the image. Mathematical descriptionTo calculate this intensity pattern, one needs to introduce some more sophisticated methods. The mathematical representation of a radial wave is given by where The absolute value of this function gives the wave amplitude, and the complex phase of the function corresponds to the phase of the wave.
Since we are for the moment only interested in the amplitude and relative phase, we can ignore any overall phase factors that are not dependent on The sum has the form of a geometric sum and the can be evaluated to give The Intensity is given by the absolute value of the complex amplitude squared Quantitative analysis of single-slit diffraction
As an example, an exact equation can now be derived for the intensity of the diffraction pattern as a function of angle in the case of single-slit diffraction. A mathematical representation of Huygens' principle can be used to start an equation. Consider a monochromatic complex plane wave If the slit lies in the x′-y′ plane, with its center at the origin, then it can be assumed that diffraction generates a complex wave ψ, traveling radially in the r direction away from the slit, and this is given by: Let (x′,y′,0) be a point inside the slit over which it is being integrated. If (x,0,z) is the location at which the intensity of the diffraction pattern is being computed, the slit extends from The distance r from the slot is: Assuming Fraunhofer diffraction will result in the conclusion It can be seen that 1/r in front of the equation is non-oscillatory, i.e. its contribution to the magnitude of the intensity is small compared to our exponential factors. Therefore, we will lose little accuracy by approximating it as z.
To make things cleaner, a placeholder 'C' is used to denote constants in the equation. It is important to keep in mind that C can contain imaginary numbers, thus the wave function will be complex. However, at the end, the ψ will be bracketed, which will eliminate any imaginary components. Now, in Fraunhoffer diffraction,
It can be noted through Euler's formula and its derivatives that
where the (unnormalized) sinc function is defined by Now, substituting in
Quantitative analysis of N-slit diffraction
Let us again start with the mathematical representation of Huygens' principle. Consider N slits in the prime plane of the equal size (a, To generalize this to N slits, we make the observation that while z and y remain constant, x′ shifts by Thus and the sum of all N contributions to the wave function is: Again noting that
Now, we can use the following identity
Substituting into our equation, we find:
We now make our k substitution as before and represent all non-oscillating constants by the I0 variable as in the 1-slit diffraction and bracket the result. Remember that This allows us to discard the tailing exponent and we have our answer: |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Diffraction_formalism". A list of authors is available in Wikipedia. |