ABSTRACT
However we have shown in Eqs.(18.29) and (18.30) that the square of the absolute value of a complex scalar wave function Ψ is a measure of the wave intensity, which is the quantity of primary interest for the theories of interference and diffraction for either mechanical or electromagnetic waves. The periodic spatial variation of intensity throughout a region of superposition of waves which originate in the same source is known as interference. Its theory is based essentially on the linearity of the wave equation, which allows a linear combination of partial waves to constitute the actual wave at any given point. Consider first the superposition of two monochromatic plane waves, having the complex representation (16.25), that is
(18.1)
Since we are usually interested in the relative variation of the intensity only, in a medium 0 ,Z and not in its absolute value, it is common practice to adopt simplified wave intensities, compared to those in Eqs.(18.29) or (18.30), which read
( ) ( )2 21 1 2 2, , ,I r t I r= Ψ = Ψ tG G (18.2) The superposition of monochromatic waves (18.1) results in an intensity
( ) ( ) ( )2 21 1 2, , ,tG ( )I r t r t r= Ψ = Ψ +ΨG G ( ) ( ) ( )2 2 *1 2 1 2, , 2Re ,r t r t r t r t,⎡ ⎤= Ψ + Ψ + Ψ Ψ⎣ ⎦G G G G ( )1 2 1 2 1 2 1 22 cosI I I I k r k r ϕ ϕ= + + ⋅ − ⋅ + −G GG G (18.3) where the last term is called the interference term. If the phase difference, given by ( )1 2 1k k r 2δ ϕ ϕ= − ⋅ + −G G G (18.4) is a constant, the two waves are said to be mutually coherent, and the interference term indicates a sinusoidal spatial variation between maxima and minima, of intensity
max 1 2 1 22 ,I I I I I m2δ π= + + = (18.5)
( )min 1 2 1 22 , 2I I I I I m 1δ π= + − = +
B A
SS 21
Figure 18.1. Interference pattern produced by two monochromatic waves originating from sources 1 2andS S
Hence, the stationary interference pattern consists of a succession of surfaces, defined by the condition Nδ π= where .12or2 += mmN It is usually produced by superposition of waves originating in two point sources , which can be assumed, for simplicity, to be of equal phase constants
21 and SS
1 2 0.ϕ ϕ− = If the two sources are far enough from a given point, at distances 1 2 1 2,r r r r>> − ,G G the incident waves at the point can be regarded as plane waves, as we shall show in Chapter 19. Under this assumption Eq.(18.4) reduces to
( ) ( )1 2 1 2 1 2or 2k k r k r r N r r N Nkπ λδ π= − ⋅ ≅ − = − = =G G G (18.6)
Both the antinodal and nodal surfaces ( mN 2= ) ( )12 += mN are defined by Eq.(18.6) to be hyperboloids of revolution. Their intersection with a plane will produce alternate nodal and antinodal lines called fringes of the form of either hyperbolas or circular rings if the plane is parallel (A) or normal (B) to the line joining the sources (Figure 18.1). The interference fringes are not observed if the phase difference 1 2ϕ ϕ− has a random variation with time, since the interference term in Eq.(18.3) is averaged out. In this event the waves are said to be mutually uncoherent and do not interfere by superposition. Mutually coherent waves that originate in the same source can be obtained by splitting a single beam, so that on recombining partial waves interference effects occur.
