2-DIMENSIONAL INTERFERENCE

The most commonly studied 2-D pattern studied is from 2 COHERENT point sources - same frequency and same phase. The wave forms can be any type; sound, light, wave-particle.

The effect can be heard with loud speakers sending the same signal at about 100Hz if the speakers are a couple of wavelengths apart ( ie about 340 /100 ~ 3m apart ) very clearly.

It can be seen using lasers and narrow slits about 100μm apart. ( The first person to carry out the light version was Thomas "Phenomenon" Young back in 1801 when he use twin pin holes to demonstrate that Newton was at least partly wrong about light - that it had a wavelike character. ( Young could speak and read Latin, Greek, French, Italian, Hebrew, Arabic and Persian, had studied literature, classics, mathematics, physics and biology by the age of 14. He was one of the translators of the Rosetta Stone in later life - the stone that allowed the first translations of Egyptian hieroglyphs.)

In the animation below, only "crests" are shown as I ran out of enthusiasm and energy to draw dotted lines for "troughs" - minima in the sin wave pattern.

Every 2 seconds, the animation shows the confocal hyperbolic set of Antinodal lines. These correspond to lines along which the circular waves from the two sources always arrive exactly in phase, crest meets crest and trough meets trough.If you carefully watch the lines you will see this.

Between are a similar set of Nodal lines where crest always meets trough so cancelling out.

The Path Difference, defined as the physical difference in distances from the sources to a point of interest (usually measured in wavelengths) is crucial to understanding this interference.

Notice that between S₁ and S₂, we will have a 1-D standing wave pattern.

Path Difference is used in all sorts of interference situations. When multiple wave paths are considered whether in thin films, single slit diffraction type interference or diffraction gratings, the key is the Path Difference.

Generally, when

PD =nλ, n= 1, 2, 3, 4, ....then Constructive Interference takes place. ( Antinodes form )

BUT if

PD = (2n-1)λ / 2, ........... then destructive interference takes place. - ( Nodes form )

( Complications for thin film interference include inversion of a wave ( π/2 phase change ) on reflection and the wavelength change due to the refractive index of a material.)

WHEN THE SOURCES ARE VERY CLOSE TOGETHER compared to the distance along an antinode THEN - we can make simplifying approximations like sinθ = tanθ - which is more or less true for angles smaller than 5⁰ - check it on your calculator if you don't believe it.

Suppose the above diagram really does represent a very distant screen, that is S₁S₂ = d <<>

P₀ is always on an antinode as it is symmetrical from S₁ and S₂ - no path difference. If we move along the screen, ie y changes, then we shall pass through nodes when PD =λ/2, 3λ/2, 5λ/2 ........ and antinodes when PD =λ, 2λ, 3λ, 4λ, ....

Where to identify the PD? We could do this at the screen which is wot we did earlier, or near the sources, which is wot we will do this time.

From the above diagram, when y is at the first antinode off axis, PD =λ and

y / L = PD / d =λ / d so λ = yd / L

IIt was with this equation that Young first estimated the wavelength of light. Naturally at first he was rubbished a bit - after all Newton was KING. ItIt

Later, Fraunhofer, Fresnel and Fourier concluded that light really was wavelike - Yah Boo Sucks to Newton - but Nature really is a funny old thing - ever heard of a particle called a photon?

PS - The distance from a node to node is called a Bandwidth W, which appears often for "y".

Diffraction Gratings