Phasors

Phasors provide a convenient way to analyze interference phenomena. A phasor is simply a two-dimensional vector that represents a complex number: the $x$-component of a phasor represents the real ( $\Re\mathfrak{e}$) part of a complex number and the $y$-component represents the imaginary ( $\Im\mathfrak{m}$) part. The polar representation of complex numbers also has a neat geometric interpretation with phasors: the angle that the phasor makes with the $x$-axis represents the phase of the complex number, and the length of the phasor represents its magnitude. To represent a sum of two or more complex numbers, you can now simply add the corresponding phasors, using the usual rules of vector addition.

Example: the sum $2 + 4e^{i\pi/3} \approx 5.3 e^{i0.714\pi}$ can be represented by the phasor diagram in Fig. 4.

Figure 4: Adding complex numbers with phasors.

We can use phasors to illustrate the sum inside the absolute value bars in Eq. (8) (p. 7 of the Class Notes ``Wave Phenomena II: Interference''.) Each phasor represents the complex amplitude of the light coming from one source. When we sum these phasors, the resultant phasor represents the complex amplitude of the superposition of light coming from all of the sources.

Example: A minimum for three equivalent narrow slits can result from either of the phasor sums in Fig. 5.

Figure: Phasor diagram for 3-slit interference.

Draw the phasor diagrams to illustrate the following cases of interference. Label magnitudes and angles. If there is more than one possibility (as in the example above), illustrate all of them.

(a)

Principal maximum for 5 narrow slits;

(b)

Minimum for 5 narrow slits ;

(c)

Secondary maximum for 4 narrow slits (don't worry about making the angles exact--an approximation is fine);

(d)

Minimum for 2 narrow slits where one is three times as wide as the other ($I_1=3I_2$).