Diffraction is the tendency of a wave emitted from a finite source or passing through a finite aperture to spread out as it propagates.  Diffraction results from the interference of an infinite number of waves emitted by a continuous distribution of source points.  According to Huygens principle every point on a wave front of light can be considered to be a secondary source of spherical wavelets.

The wave fronts of a spherical wave are spherical surfaces surrounding a point source.  The wave vector k points away from the point source along the radius of the sphere at any point.  If the point source is located at the origin, then k is always parallel to the position vector r, k || r, and k×r = k r.  Here k = 2p/l is the wave number and r is just the scalar distance along any radial direction.

The area of a wave front with radius r is 4pr².  Since energy is conserved, the intensity must decrease with r as 1/r².  Since the intensity is proportional to the square of the amplitude, the amplitude must decrease with r as 1/r, and we have E(r) = (A/r)cos(kr-wt), with A being a constant.

Maxwell’s equation can be used to formulate an exact description of the propagation of a light wave through an optical system and the space around it.  This is rarely done.  Two standard approximations to the exact formulation of wave propagation and diffraction are the Fraunhofer and Fresnel approximations.  These approximations ignore the vector nature of the electric field.

The Fraunhofer regime is the far-field regime.  Very far from a point source the wave fronts are essentially plane waves.  The Fraunhofer approximation is only valid when the source, aperture, and detector are all very far apart or when lenses are used to convert spherical waves into plane waves.

The Fresnel regime is the near-field regime.  In this regime the wave fronts are curved, and their mathematical description is more involved.

The single slit

Assume light from a distant source passes through a narrow slit.  What do we observe on a distant screen?

According to the Huygen-Fresnel principle, the total field at a point y on the screen is the superposition of wave fields from an infinite number of point sources in the aperture region.  Each point s on the wave front inside the aperture ( –a/2 £ s £ a/2) is the source of a spherical wave.  A distance r from the point s the electric field is due to this point sources is 

dE = (A_sds/r)cos(kr-wt).

If r₀ is the distance from the point s = 0 on the optical axis to a point y on the screen, then the contribution dE to the total amplitude on the screen from the point at s = 0 is

dE(y) = (A_sds/r₀)cos(kr₀-wt).

Here A_s/r₀ is the amplitude per unit width and ds is the infinitely small width of a point source.  For off-axis points for which s ¹ 0, the distance is longer or shorter than r₀ by an amount D.

The contribution dE(y) to the total amplitude on the screen from an off-axis point (s ¹ 0) is

dE(y) = (A_sds/(r₀+D(s)))cos(k(r₀+D(s))-wt).

To find the total amplitude E(y) we have to add up the contributions from all points on the aperture.  Because there are an infinite number of points, the sum becomes an integral.

.

We define sinq = D/s.  Since r₀ >> D, we approximate 1/(r₀+D) with 1/r₀.  However we cannot drop the D inside the cosine function, since kD(s) is not necessarily much smaller than 2p.

We then have

.

Using

,

integration then yields

.

The function sin(x)/x = sinc(x) is called the sinc function.

The intensity is proportional to the square of the field,

.

Since the square of a cosine function averages to ½, the time-averaged intensity is given by

,

where <I₀> µ (1/2)(A_sa/r₀)² and k = 2p/l.

The time-averaged intensity has a peak in the center with smaller fringes on the sides.

For small angles we may approximate sinq @ q.  Then the first zeros on the sides of the central peak occur when

pasinq/l @ paq/l = p, or q = l/a.

On the screen we see a pattern similar to that shown in the figure below.

The positions of all maxima and minima in the diffraction pattern from a single slit can also be found from the following simple arguments.

When light passes through a single slit whose width w is on the order of the wavelength of the light, then we observe a single slit diffraction pattern.  Huygen's principle tells us that each part of the slit can be thought of as an emitter of waves.  All these waves interfere to produce the diffraction pattern. Consider a slit of width w as shown in the diagram below.

For light leaving the slit in a particular direction, we may have destructive interference between the ray at the top edge (ray 1)and the middle ray (ray 5).  If these two rays interfere destructively, so do rays 2 and 6, 3 and 7, and 4 and 8.  In effect, light from one half of the opening interferes destructively and cancels out light from the other half.  Ray 1 and ray 5 are half a wavelength out of phase if ray 5 must travel 1/2 wavelength further than ray 1.  We need

(w/2)sinq = l/2 or wsinq = l

for destructive interference to produce the first dark fringe.  Other dark fringes in the diffraction pattern produced by a single slit are found at angles q for which

wsinq = ml.

If the interference pattern is viewed on a screen a distance L from the slits, then the wavelength can be found from the spacing of the fringes.  We approximately have

l = zw/(mL),

where z is the distance from the center of the interference pattern to the mth dark line in the pattern.  That applies as long as the angle q is small, i.e. as long as z is small compared to L

Problem:

Diffraction of light through a rectangular aperture is a rather straightforward extension of 1-dimensional diffraction from a slit, as shown in the diagram below.

A circular aperture is qualitatively similar, but an accurate quantitative treatment of the pattern requires more complicated mathematics using "Bessel Functions".  The main result is shown in the diagram below.  The intensity pattern is called the "Airy Disk".  The first minimum occurs at q, where q = 1.22l/D, where D is the diameter of the aperture.  On a screen a distance L >> D from the aperture it is seen at a radial distance r' = 1.22lL/D from the center of the pattern.

Producing a laser beam is an attempt to confine the light in the directions transverse to the direction of propagation.  The light will spread out in the same way it does after passing through an aperture. 

The angle through which the light spreads is approximately q @ l/a(0).  Therefore a(z) @ lz/a(0).  Because the laser beam diameter is typically much larger the wavelength of light, or a(0) >> l, q is quite small.  Consider a HeNe laser, for which l = 633 nm and a beam diameter is 0.633 mm.  Then q = 10^-3 radian = 1 milliradian.  The beam must propagate ~ 6.33 m before the diameter increases by a factor of 10.

In geometrical optics we assume that an ideal, aberration-free lens focuses parallel rays to a single point one focal length away from the lens.  But the lens itself acts like an aperture with diameter D for the incident light.  The light passing through the lens therefore spread out.  This yields a blurred spot at the focal point.  Light near the focal point exhibits an Airy Disc pattern.  The size of the Airy Disc is determined by the focal length f and diameter D of the lens.  The radius r of the Airy Disc at the focal point of a lens is given by r =1.22lf/D.

If all ray aberrations in an optical system can be eliminated, such that all of the rays leaving a given object point land inside of the Airy Disc associated with the corresponding image point,  then we have a diffraction-limited optical system. This is the absolute best we can do for an optical system that has lenses with finite diameters.

Resolving power

The resolving power of an optical instrument is its ability to separate the images of two objects, which are close together.  Some binary stars in the sky look like one single star when viewed with the naked eye, but the images of the two stars are clearly resolved when viewed with a telescope.  
Why? 
The merging of the images in the eye is caused by diffraction.

If you look at a far-away object, then the image of the object will form a diffraction pattern on your retina.  For two far-away objects, separated by a small angle q, the diffraction patterns will overlap.  You are able to resolve the two objects as long as the central maxima of the two diffraction patterns do not overlap.  The two images are just resolved when one central maximum falls onto the first minimum of the other diffraction pattern.  This is known as the Rayleigh criterion.  If the two central maxima overlap the two objects look like one

The width of the central maximum in a diffraction pattern depends on the size of the aperture, (i.e. the size of the slit).  The aperture of your eye is your pupil.  A telescope has a much larger aperture, and therefore has a greater resolving power.  The minimum angular separation of two objects which can just be resolved is given by q_min = 1.22l/D, where D is the diameter of the aperture.  The factor of 1.22 applies to circular apertures like the pupil of your eye or the apertures in telescopes and cameras.

The closer you are to two objects, the greater is the angular separation between them.  Up close, two objects are easily resolved.  As your distance from the objects increases, their images become less well resolved and eventually merge into one image.

Problem:

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