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Multiple Wave Interference

Multiple Wave Interference with Decreasing Amplitude

We will turn our attention now to a slight modification of the previous multiwave interference. We will introduce a decreasing amplitude of the individual waves. The first wave shall have an amplitude \(U_1=\sqrt{I_0}\). The next wave, however, will not only be phase shifted but also have a smaller amplitude.

\[U_2=h U_1\]

where \(h=re^{i\phi}\) with \(|h|=r<1\). \(r\) can be regarded as a reflection coefficient, which deminishes the amplitude of the incident wave. According to that the intensity is reduced by

\[I_2=|U_2|^2=|h U_1|^2=r^2 I_1\]

The intensity of the incident wave is multiplied by a factor \(r^2\), while the amplitude is multiplied by \(r\). Note that the phase factor \(e^{i\phi}\) is removed when taking the square of this complex number.

Intensity at Boundaries

The amplitude of the reflected wave is diminished by a factor \(r\le 1\), which is called the reflection coefficient. The intensity is diminished by a factor \(R=|r|^2\le1\), which is the reflectance.

In the absence of absorption, reflectance \(R\) and transmittance \(T\) add to one due to energy conservation.

\[R+T=1\]

Consequently, the third wave would be now \(U_3=hU_2=h^2U_1\). The total amplitude is thus

\[U=U_1+U_2+U_3+\ldots+U_M = \sqrt{I_0}(1+h+h^2+\ldots)\]

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Fig.: Multiple wave interefence with decreasing amplitudes. Left: Phase construction of a multiwave intereference with M equal amplitude waves. Right: Intensity distribution obtained as a function of the phase shift \(\phi\).

This yields again

\[U=\sqrt{I_0}\frac{(1-h^M)}{1-h}=\frac{\sqrt{I_0}}{1-r e^{i\phi}}\]

Calculating the intensity of the waves is giving

\[I=|U|^2=\frac{I_{0}}{|1-re^{i\phi}|^2}=\frac{I_0}{(1-r)^2+4r\sin^2(\phi/2)}\]

which is also known as the Airy function. This function can be further simplified by the following abbrevations

\[I_{\rm max}=\frac{I_0}{(1-r)^2}\]

and

\[\mathcal{F}=\frac{\pi \sqrt{r}}{1-r}\]

where the latter is called the Finesse. With those abbrevations, we obtain

\[I=\frac{I_{\rm max}}{1+4\left(\frac{\mathcal{F}}{\pi}\right)\sin^{2}(\phi/2)}\]

for the interference of multiple waves with decreasing amplitude.

This intensity distribution has a different shape than the one we obtained for multiple waves with the same amplitude.

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Fig.: Multiple wave interference with decreasing amplitude. The graph shows the intensity distribution over the phase angle \(\phi\) for different values of the Finesse \(\mathcal{F}\).

We clearly observe that with increasing Finesse the intensity maxima, which occur at multiples fo \(\pi\) get much narrower. In addition the regions between the maxima show better contrast and fopr higher Finesse we get complete destructive interference.