The single slit

If the wavelengths of the light become comparable to the dimensions of the equipment, then we study optical phenomena using the classical theory of radiation, or wave optics.  Wave optics contains all of ray optics, but the mathematical treatment is much more involved. 

Diffraction and interference are phenomena observed with all waves.  Diffraction can only be observed with waves traveling in two or three dimensions.

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 in two or three dimensions.  Huygens' principle lets us treat wave propagation by considering every point on a wave front to be a secondary source of spherical wavelets.  These wavelets propagate outward with the characteristic speed of the wave.  The wavelets emitted by all points on the wave front interfere with each other to produce the traveling wave.  Huygens' principle also holds for electromagnetic waves.  When studying the propagation of light, we can replace any wave front by a collection of sources distributed uniformly over the wave front, radiating in phase.

When light passes through a small opening, comparable in size to the wavelength λ of the light, in an otherwise opaque obstacle, the wave front on the other side of the opening resembles the wave front shown on the right.

The light spreads around the edges of the obstacle.  This is the phenomenon of diffraction.

When light passes through a single slit whose width w is on the order of the wavelength of the light, then we can observe a single slit diffraction pattern on a screen that is a distance L >> w away from the slit.  The intensity is a function of angle.  Huygens' 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.  Where crest meets crest we have constructive interference and where crest meets trough we have destructive interference.

Maxwell's equations define classical electromagnetism, and therefore classical wave optics.  Many types of active optical phenomena and devices are excluded.  A laser is an example of an active optical device.  But classical optics is used to design the resonator cavity to confine the optical fields within the active laser medium.

Maxwell’s equations lead to a vector formulation of diffraction i.e. to a mathematical formulation of Huygens' graphical construction.  In  the old days of optics, scalar equations similar to the vector equations derived from Maxwell’s equations were derived in various ad hoc ways.  An example is the Fresnel diffraction theory.  It derives the Fresnel diffraction integral, which in principle, can be evaluated for any source distribution in front of any aperture to predict the intensity distribution of the diffracted light behind the aperture at any distance.  However, the integral can most often only evaluated numerically.  This must be done when the sources are located near the apertures and/or the diffraction pattern close to the apertures is to be calculated.  In this regime the wave fronts are curved at the apertures.

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Very far from a point source the wave fronts are essentially plane waves.  This is called the Fraunhofer regime, and the diffraction pattern is called Fraunhofer diffraction.  The Fraunhofer approximation of the Fresnel diffraction theory 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.  Being very far apart means that the distances between source, aperture, and detector must be much greater than the width of the aperture.

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Fraunhofer approximation

Assume light from a distant source passes through a narrow slit as shown on the figure below.  Assume the incident wave can be treated as a plane wave E(r,t) = k Ecos(kx - ωt + φ).  The polarization of this wave perpendicular to the plane of the figure.  Let the slit be very long.  Let us concentrate on the region near the middle of the slit and neglect variations of the transmitted wave perpendicular to the plane of the figure.  (We are reducing the problem from a 3D to a 2D problem.)  What do we observe on a distant screen?

According to the Huygens-Fresnel principle, the total field at a point y = d*tanθ on a screen a large distance d from the slit is the superposition of radiation fields from an infinite number of point sources in the aperture region.  Each section on the wave front between s and s + ds inside the aperture (-a/2 ≤ s ≤ a/2) is the source of a spherical wave.  A large distance r from the point source between s and s + ds the radiation field is due to the sources is

dE = (A_sds/r)cos(kr - ωt).

Here A_sds/r the field at a distance r from the point source of infinitesimally small width ds.

If r₀ = d 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 = d*tanθ) = (A_sds/r₀)cos(kr₀ - ωt).

For off-axis points for which s ≠ 0, the distance to the screen is longer or shorter than r₀ by an amount Δ.

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

dE(y) = (A_sds/(r₀ + Δ(s))) cos(k(r₀ + Δ(s)) - ωt).

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.

E(y) = ∫_-a/2^+a/2(A/(r₀ + Δ(s)))cos(k(r₀ + Δ(s)) - ωt)ds.

We define sinθ = Δ/s.  Since r₀ >> Δ, we approximate 1/(r₀ + Δ) with 1/r₀.  However we cannot drop the Δ inside the cosine function, since kΔ(s) is not necessarily much smaller than 2π.  We then have

E(y) = (A_s/r₀)∫_-a/2^+a/2cos[(ksinθ)s + (kr₀ - ωt)]ds.

Using ∫cos(ax + b)dx = (1/a)sin(ax + b) the integration yields

E(y) = (A_sa/r₀)cos(kr₀ - ωt)(sin(ka(sinθ)/2)/(ka(sinθ)/2).

or, inserting k = 2π/λ,

E(y) = (A_sa/r₀)cos(kr₀ - ωt)(sin(πa(sinθ)/λ)/(πa(sinθ)/λ).

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

The intensity is proportional to the square of the field,  I(y) E²(y).  Since the square of a cosine function averages to ˝, the time-averaged intensity is given by

<I(y)> = <I₀>sin²(πa(sinθ)/λ)/(πa(sinθ)/λ)²,

where <I₀> is the average intensity at the center.

The time-averaged intensity has a peak in the center with smaller fringes on the sides.  For small angles we may approximate sinθ ~ θ.  Then the first zeros on the sides of the central peak occur when πasinθ/λ ~ πaθ/λ = π, or θ = λ/a.

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

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

A circular aperture is qualitatively similar, but an accurate quantitative treatment of the pattern requires more complicated mathematics.  The intensity pattern is called the "Airy Disk".  The main features are shown in the diagram below.  The first minimum occurs at an angle θ = 1.22 λ/D, where D is the diameter of the aperture.  On a screen a distance L >> D from the aperture the minimum is seen at a radial distance r' = 1.22 λL/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.

Assume that at z = 0 the diameter of a laser beam is restricted to a(0).  The angle through which the light spreads is approximately θ ≈ λ/a(0).  (For back-of-the-envelope calculations we often ignore the factor of 1.22.)  Because the laser beam diameter is typically much larger the wavelength of light, or a(0) >> λ, θ is quite small.  At a large distance z the diameter of the beam will have increased to  a(z) ≈ z*2θ.  Consider a HeNe laser, for which λ = 633 nm with a beam waist of ~ 0.6 mm.  Then θ ~ 10^-3 rad = 1 millirad.  The beam must propagate ~ 3 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 = 1.22 λf/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.

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.

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 θ, 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 θ_min = 1.22 λ/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.

When light passes through an aperture with diameter D, then diffraction limits the resolution to θ = 1.22λ/D.  If the angular separation of two sources is less than θ, they cannot be resolved.  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:

A spy satellite travels at a distance of 50 km above Earth's surface.  How large must the lens be so that it can resolve objects with a size of 2 mm and thus read a newspaper?  Assume the light has a wavelength of 400 nm. 

Solution:

• Reasoning:
  The minimum angular separation of two points which can just be resolved by an optical instrument is given by θ_min = 1.22 λ/D, where D is the diameter of the aperture of the instrument. 
• Details of the calculation:
  Diffraction limits the resolution according to θ = 1.22 λ/D = y/L.  Here the height of the object to be resolved is y and the distance to the object is L.  Solving for D we find D = 12.2 m.

AI Study Tip:

Example prompt:  'Explain the transition of an intensity pattern as we move from a single slit to a circular aperture using the Airy Disk formula.  Use a metaphorical analogy (like ripples in a pond or sound through a doorway) to explain why the central maximum contains most of the energy, and then describe how the Rayleigh Criterion defines the 'limit of resolution' for two distant stars being imaged by a telescope.'