What is Optical path and phase? — Physics

Definition 18.2 University Physics — Year 2 · Chapter 18 — The Scalar Model of Light

In a medium of index $n$ the wave travels at $c/n$ and its wavelength is $\lambda_0/n$ . The optical path along a curve from $A$ to $B$ is

(AB) = \int_A^Bn\,\dd s ,

and the phase accumulated by a monochromatic wave travelling along a ray from $A$ to $B$ is

\varphi(B) - \varphi(A) = \frac{2\pi}{\lambda_0}\,(AB) = \omega\,\tau_{AB} ,

$\tau_{AB}$ being the travel time. A wavefront is a surface of equal phase; in an isotropic medium the rays of geometrical optics are perpendicular to the wavefronts (Malus–Dupin theorem), and the optical path between two wavefronts is the same along every ray.

Left: a thin lens turns plane wavefronts into spherical ones converging on the focus — all optical paths to F' are equal. Right: inside glass the wavelength shrinks to _0/n and the phase advances faster: the optical path counts the geometric length n times. — Left: a thin lens turns plane wavefronts into spherical ones converging on the focus — all optical paths to $F'$ are equal. Right: inside glass the wavelength shrinks to $\lambda_0/n$ and the phase advances faster: the optical path counts the geometric length $n$ times.

Examples

Example 18.3 (Stigmatism; the lens as a phase plate)

A point source $A$ imaged by a perfect (stigmatic) instrument at $A'$ : all the rays from $A$ to $A'$ have the same optical path — the waves arriving along every ray are in phase at $A'$ , which is why they pile up into a bright point there. For a thin lens of focal length $f$ this is a statement about phase: a plane wave arriving along the axis must leave as a spherical wave converging to the focus $F'$ ; since a spherical wave centred at distance $f$ has, at distance $r$ from the axis, the phase $2\pi r^2/2\lambda f$ ahead of its axial value (paraxial approximation, $\sqrt{f^2 + r^2} \approx f + r^2/2f$ ), the lens must retard the wave by $\varphi_L(r) = -\pi r^2/\lambda f$ : it is thicker at the centre by exactly the amount that makes all optical paths to $F'$ equal. A lens is a phase plate; a hologram or a liquid-crystal display that imposes the same phase map is a lens.

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Examples

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