Physics · Ch 6 — Optics

Phase Difference and Path Difference

6.10.1

Phase Difference and Path Difference

Phase is the angular position of vibration of a wave. In the path of a travelling wave, one full wavelength $\lambda$ corresponds to a phase change of exactly $2\pi$ , so a general path difference $\delta$ corresponds to a phase difference $\phi$ via $\phi=(2\pi/\lambda)\delta$ (or $\delta=(\lambda/2\pi)\phi$ ). For constructive interference, the phase difference must be $\phi=0,2\pi,4\pi,\ldots$ , so the path difference must be $\delta=n\lambda$ (integer multiples of $\lambda$ ), $n=0,1,2,\ldots$ . For destructive interference, the phase difference must be $\phi=\pi,3\pi,5\pi,\ldots$ , so the …

Figure 6.53Path difference and phase difference

What this figure shows. A single wavelength of a travelling wave is marked out along its direction of propagation, spanning a phase change of exactly 2pi radians over that one wavelength lambda. This one-to-one correspondence -- one full wavelength of path corresponds to one full cycle (2pi) of phase -- is the direct basis for the conversion formula phi = (2pi/lambda) delta linking any given path difference delta to its eq …

Misc Example 6.27Phase difference for a 3 mm path difference at 450 nm wavelength

Worked out. Light of wavelength 450 nm (450 times 10 to the -9 m) has a path difference of 3 mm (3 times 10 to the -3 m) introduced between two interfering beams. Substituting into phi = (2pi/lambda) delta gives phi = 2pi times (3 times 10 to the -3)/(450 times 10 to the -9) = 2pi times 6667 = 75 times 10 to the 6 pi radians. Because this phase difference works out to an exact multiple of pi (a huge one, since the path difference is thousands of wavelengths), whether the resulting interference at that point is constructive or destructive depends only on whether 75 times 10 to the 6 comes out even (constructive) or odd (destructive) when the underlying multiple of lambda is examined directly -- illustrating how even a very small physical path difference corresponds to an en …

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