A Michelson interferometer works by splitting a beam of light into two paths, reflecting the beams back, and then recombining them. When the path length of one arm changes, it affects the interference pattern. The condition for constructive interference (bright fringes) and destructive interference (dark fringes) is related to the optical path difference (OPD) between the two beams.
The fringe pattern will shift from bright to dark (or vice versa) when there is a change in optical path length equal to half the wavelength of light. This is because:
- A bright fringe occurs when the OPD is a multiple of the wavelength, given by the condition: \[ \Delta L = m \lambda \quad \text{(for constructive interference)} \] where \( m \) is an integer.
- A dark fringe occurs when the OPD is an odd multiple of half the wavelength, given by: \[ \Delta L = (m + \frac{1}{2}) \lambda \quad \text{(for destructive interference)} \]
To invert the fringe pattern from bright to dark, the path length needs to change by \( \frac{1}{2} \lambda \). Therefore, if \( \lambda = 465 \) nm, the minimum distance that the mirror must move (which changes the optical path length) is:
\[ \Delta L = \frac{1}{2} \lambda = \frac{1}{2} \cdot 465 \text{ nm} = 232.5 \text{ nm} \]
Since the movement of the mirror changes the length of the path for that beam by twice the actual distance moved (because it travels to the mirror and back), the actual movement of the mirror can be calculated as:
\[ \text{Minimum mirror movement} = \frac{\Delta L}{2} = \frac{232.5 \text{ nm}}{2} = 116.25 \text{ nm}. \]
Thus, the minimum distance that the device could have moved to cause the inversion of the fringe pattern is:
\[ \boxed{116.25 , \text{nm}}. \]
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