Optical Path Length and Phase

The optical path length (OPL) is the equivalent distance light would travel in vacuum to accumulate the same phase as it does traveling a physical distance $d$ through a medium of refractive index $n$:

$\text{OPL} = n \cdot d$

Why OPL Matters

Inside a medium, light travels slower ($v = c/n$) and its wavelength shortens ($\lambda_n = \lambda_0/n$). But the phase accumulated per meter of wavelength is the same as in vacuum. The OPL captures the total phase accumulation in vacuum-equivalent meters.

OPL is the foundation of:

• Interference calculations — path difference in OPL determines constructive or destructive interference
• Optical coherence — coherence length in OPL units
• Wavefront engineering — spatial light modulators and adaptive optics manipulate OPL

Phase Difference

As light travels through a medium, the accumulated phase is:

$\phi = \frac{2\pi}{\lambda_0} \cdot \text{OPL} = \frac{2\pi n d}{\lambda_0}$

Where $\lambda_0$ is the vacuum wavelength.

When two beams with OPL difference $\Delta$ interfere:

• Constructive if $\Delta = m\lambda_0$ ($m = 0, 1, 2, \ldots$)
• Destructive if $\Delta = (m + \frac{1}{2})\lambda_0$

Example: Light ($\lambda_0 = 550\,\text{nm}$) through glass ($n = 1.5$, $d = 1\,\text{cm} = 0.01\,\text{m}$):

$\text{OPL} = 1.5 \times 0.01 = 0.015\,\text{m}$

$\phi = \frac{2\pi \times 0.015}{550 \times 10^{-9}} \approx 171{,}360\,\text{rad}$

That is roughly 27,000 full oscillations — which is why the slightest path length difference matters in interferometry.

Optical Path Difference

When comparing two paths, the optical path difference (OPD) determines the interference outcome:

$\text{OPD} = n_1 d_1 - n_2 d_2$

Your Task

Implement the OPL and phase difference calculations.