PinkysBrain wrote:Can you do it again with full Maxwell equations?
Well, in a sense, I was. The problem is that the question, as is, is very difficult because it depends so much on the actual conditions. Fraunhofer gives a pretty good expression for the conditions under which Fraunhofer is valid. That is (roughly), Fraunhofer:
d > a^2/lambda
where d = source-aperture distance or aperture screen distance (which ever is smaller)
a = aperture diameter
Of course, if d is infinite the Fraunhofer condition will always be satisfied, regardless of lambda or the aperture diameter. If the condition is not satisfied and you have to use Fresnel approximations, perhaps the best way to tackle it is either by the Cornu spiral method or using the Fresnel-Kirchoff integral. The former is still approximate because the phase variations that make up the Cornu spiral may not be smooth, ie the "spiral" may not be a spiral at all. If you actually used the real phase variations, you'd probably end up with a Fresnel Kirchoff integral anyway. If you use the Fresnel-Kirchoff formulation, you need an "aperture function". This means that the aperture function will have to be idealised; so that, for a series of slits, the aperture function could be a series of delta functions, but real slits are not delta functions - they have rough edges. Now, you need to place boundary conditions on the integral, which makes it even more complicated. The perfect situation would be to model a real diffraction screen by convolving the delta functions with some sort of "edge function" and then using this in the Fresnel Kirchoff integral. The validity of this would depend on how accurate your "edge function" would be.
If now you replaced the dielectric material of the diffracting screen with a conducting screen, you'd have to take into the currents generated in the the conducting medium. You can account for the the currents by approximating an idealised conductor (infinite number of free electrons). If you did this, you'd have to assume that all free charges were on the surface. Then, using Maxwell, you'd find the E field for both perpendicular polarisation and transverse polarisation. This would give you the currents on the surface of the conductor. However, the currents themselves would not be uniform and so the expressions for the currents would involve a Fourier Transform (They don't have to. You can always use an integral representation with any set of basis functions, a la QM). In the end, the currents, which create the new diffracted fields, would be complex and you'd have to do a complex integration. This means specifying a path in i-space and assuming certain infinities (to get the residues).
As you can see, finding the diffraction limit of a real set of diffracting elements, without the assumptions of Fraunhofer, Fresnel or Kirchoff is bloody difficult! It can be done, but you'd have to consult journals of Theoretical Physics to get the details. In this sense, you may be right that Fresnel and Fraunhofer break down without exact boundary conditions.
On the other hand, the Fraunhofer approximations works well for most real gratings. If you're diffracting a raw HeNe laser beam (or even a narrow line from a cylindrical lens), then the source aperture is the diameter (or width) of the beam, let's say it's about 1mm. Lambda is 633 x 10^(-9). So for Fraunhofer to be valid, the screen where the image is focused has to be a^2/lambda = 1.5 m or about 8 or 9 feet. Not unreasonable in most situations. The diffraction limit under these conditions would be the width of the diffracted maxima, since the diffraction slit is assumed to be infinitely thin and composed of an infinite number of electrons all perfectly bound. In reality, broadening will occur because a real aperture has a finite thickness and there will also be some unbound electrons - nothing is purely a condustor or purely a die-electric . Thus collision effects of electrons within the aperture, Doppler broadening due to thermal effects (I assume you're not carrying out this experiment in absolute zero! Joking! Joking!) and currents generated by the E-field for both polarisations states will broaden the pattern. In addition, the assumption is that the electron oscillators are pure harmonic oscillators. In reality, inter-atomic forces do not allow precise harmonic oscillation. The electron is assumed to oscillate harmonically because the dip in the Van der Waals potential is close to the harmonic potential near the origin. So, the exponentials used in the calculation themselves are approximated due to the assumptions of harmonic potentials that don't really exist.
With all this in mind, let me start with the fact that Maxwell's equations can be reduced to the wave equation:
d/dx(dE/dx) = 1/c^2*d/dt(dE/dt) This is, of course partial, but I have no way of typing partial derivatives.
assuming a solution of
E = f(x)g(t)
gives a solution for the wave equation of
E = E(0)exp(i(k.r - wt)
giving rise to the description of the wave as an harmonic oscillator in time and space. The Principal of Superposition allows us to combine several waves from several sources by simply adding up the E-fields. This solution comes directly from Maxwell - albeit with no current sources and in a vaccuum.
This is a single wave of infinite extent with a single frequency, which is impossible. So, again, the actual wave is a superposition of several components giving a wave packet. You have to integrate over all the frequency components (or wave vector components) in the wave packet to calculate the real propagation of a real wave. In a real medium, you also have dispersion caused by local (bound) current sources, which needs to be taken into account.
However, proceeding in our happy, happy world of total perfection, the diffracted beam is the superposition of a series of exponentials, each coming from one electron in a thin, perfect dielectric, perfectly harmonic mode:
E = E(0){exp(i(kr_1 - wt)) + exp(i(kr_2 - wt)) + ... + exp(i(kr_N - wt))}
where all the r's are the distance from the electron to the point of observation.
This can be expressed:
E = E(0)*exp(i(kr_1-wt_))* {1 + exp(i*delta) + exp(i*delta)^2 + exp(i*delta)^3 + ... exp(i*delta)^(N-1)}
This is a geometric series which sums to:
(exp(i*delta*N) - 1)/(exp(i*delta - 1) = exp(i*(N-1)*delta/2)*(sin(N*delta/2))/sin(delta/2)
Using exp(i*x) = cos(x) + i*sin(x) and exp(-i*x) = cos(x) - i*sin(x)
If you plug this back into the expression for for the E field, you'll eventually get the expression for the diffraction pattern of a thin line of electron oscillators. In this way, the diffraction field is directly obtained from Maxwell. You might argue that it it's possible to get a real diffracted field from a real grating by starting right back from Maxwell, and this would be true. However, as i mentioned above, this would lead to Fresnel-Kirchoff integrals and complex-plane integration, all of which can be traced back to Maxwell's solution of exp(i*(k.r - wt)) and the principal of superpostion; except that the superposition is a much more complex one than simple addition of harmonic functions.
At any rate, as mentioned previously, this result can then be used to derive the expression for diffraction from a series of parallel slits giving the expression I gave earlier:
I(theta) = I(0)*sinc^2(q)*(sin^2(N*p)/sin^2(p)
In this case, the diffraction limited linewidth is given by the minima of the function for the slits which occur whenever (sin^2(N*p)/sin^2(p) = 0, or when
p = +/-pi/N, +/- 2*pi/N, +/- 3*pi/N etc.
So, the width of the central maxima, the diffraction limited linewidth, is 2*pi/N
If it helps, I could give you the derivation of the diameter of the Airy disc, which is usually considered to be the diffraction limited spot from a lens. The theory is developed in the same way (superposition of ideal harmonic oscillators froma thin disc of electrons in an aperture, or lens), but now a circular coordinate system is used. In this case, the diffraction pattern is a Bessel function and is:
I(theta) = I(0){2J(1)(k*a*sin(theta))/k*a*sin(theta)
where J(1) is a Bessel function of the first order.
This shows a central peak surrounded by a dark ring with subsidiary peaks of lower intensity. The radius of the central peak (the diffraction limit of a lens) is given by the zeroes of the Bessel function. The first zero of the bessel function J(1)(u) occurs when u = 3.83 (from tables). If the radius of the first disk (the diffraction limit, remember) is r, then
sin(theta) = r/R
where R = distance to screen from lens. And so:
k*a*sin(theta))/k*a*sin(theta) = k*a*r/R = 3.83
Thus, the radius of the first zone of the Airy disc is
r = 1.22 lambda*R/2*a
For a lens, R = focal length and a = radius of lens, so the diffraction limitted radius of a lens is:
r = 1.22 f*lambda/D
where D = diameter of lens.
As you can see, this is derived in exactly the same way (except that it's circular) and we gloss over the details of a real lens made of a real material with rough edges (and scratches all over it, if my lenses are representaticve!)