Module 5.1 Newton’s Rings Interference (Reflected & Transmitted Systems)

1. Experimental Optical Bench Setup & Ray Paths

The experimental arrangement designed by Sir Isaac Newton and perfected for precision interferometry consists of the following components:

Figure 1: Complete laboratory optical bench setup showing extended monochromatic source, condensing lens, 45° glass plate, plano-convex lens, optical flat, and traveling microscope eyepiece.

• Extended Monochromatic Source ($S$): Typically a Sodium vapor lamp emitting intense yellow light at $\lambda = 589.3\text{ nm}$ (mean of $D_1$ and $D_2$ doublet). An extended broad source is essential so that rays enter the pupil over a wide range of angles, ensuring a broad field of view.
• Condensing Lens ($L_1$): Collimates diverging rays into a broad parallel horizontal beam directed toward the beam splitter.
• Glass Plate ($G$) at $45^\circ$: Inclined at $45^\circ$ to the vertical beam axis. It partially reflects 50% of the light normally downward onto the lens-plate system ($i = 0, \cos r = 1$), eliminating angular skew. Reflected interference rays from the film pass upward through $G$ directly into the microscope.
• Plano-Convex Lens ($L$) & Plane Glass Plate ($P$): The lens has an extremely large radius of curvature ($R \sim 1\text{ to }3\text{ m}$) so the air gap thickens very gradually. The glass plate $P$ is an optical flat with surface irregularities less than $\lambda/10$.
• Traveling Microscope: Mounted vertically above plate $G$, fitted with a crosswire eyepiece and micrometer screw gauge to measure ring diameters $D_m$ with sub-millimeter precision.

2. Thin Film Geometry & Stokes' Phase Jump

Consider a sphere of radius $R$ forming the curved surface of the lens. At radial distance $r$ from the central contact point $O$, let the film thickness be $t(r)$.

Figure 2: (Left) Geometry of the spherical air film showing $t \approx r^2 / (2R)$. (Right) High-magnification ray splitting for Reflected (Stokes' $\pi$ phase jump) and Transmitted rays.

From the right-angled triangle formed with the center of curvature $C$:

Since $t \ll R$ (film thickness is typically a few micrometers while $R \sim 1\text{ m}$), the second-order term $t^2$ is negligible:

If a tiny dust particle or spacer separates the surfaces by central distance $t_0$, the total thickness is $t(r) = t_0 + \frac{r^2}{2R}$.

3. Reflected Light System: Why the Center is Pitch DARK

In the reflected system, the incident ray strikes the lens-film boundary and divides into two coherent beams:

• Ray 1: Reflected at the lower curved surface of the lens (glass of index $n_g \approx 1.5$ to air/film of index $\mu \approx 1.0$). This reflection occurs at a rarer medium interface, producing zero phase shift ($\Delta\phi_1 = 0$).
• Ray 2: Transmitted into the film, traverses distance $t$, reflects at the flat glass plate (air/film of index $\mu$ to glass of index $n_g$). This reflection occurs at a denser medium boundary. By Stokes' relations (electromagnetic boundary conditions), an abrupt phase jump of $\pi$ radians (equivalent to an optical path difference of $\lambda/2$) occurs!

Under normal incidence ($\cos r = 1$), the total effective optical path difference for reflected light is:

4. Transmitted Light System: Complementary Bright Center

In the transmitted light arrangement, interference occurs between:

• Ray 1': Directly transmitted through the lens, film, and glass plate without internal reflection.
• Ray 2': Suffers two internal reflections inside the film (first at the flat plate, then at the curved lens) before emerging downward. Because both internal reflections occur at identical index steps (or because $2 \times \pi = 2\pi \equiv 0$), there is no net phase jump of $\lambda/2$!

Therefore, the effective optical path difference for transmitted rays is simply:

5. Strict Complementarity & Laboratory Determination of $\lambda$ and $\mu$

By conservation of energy, the sum of reflected and transmitted intensities satisfies $I_{\text{ref}}(r) + I_{\text{trans}}(r) = I_0$. The two fringe patterns are strictly complementary:

Figure 3: (Left) Visual comparison showing exact spatial complementarity between Reflected (dark center) and Transmitted (bright center) rings. (Right) Linear calibration graph of $D_m^2$ vs $m$ for air ($\mu=1$) and water ($\mu=1.33$).

Why Fringes Get Closer Together (Fringe Compression)

The fringe width $\beta_m$ between consecutive rings is given by the difference in consecutive radii:

As the ring order $m$ increases, $\beta_m$ decreases inversely with $\sqrt{m}$. Thus, Newton's rings are widely spaced near the center and become progressively crowded together as $r$ increases.

Determination of Unknown Wavelength $\lambda$

Let $D_m$ and $D_{m+p}$ be the diameters of the $m$-th and $(m+p)$-th dark rings in air ($\mu = 1$). Then:

Crucial Advantage: Any zero error or central air gap $t_0$ (caused by dust or lack of perfect contact) enters equally into $D_{m+p}^2$ and $D_m^2$ and cancels out completely in the difference $(D_{m+p}^2 - D_m^2)$!

Determination of Liquid Refractive Index $\mu$

When a drop of liquid (e.g., water $\mu = 1.33$ or oil $\mu = 1.5$) is placed between the lens and plate:

Since $\mu > 1$, introducing a liquid contracts the rings, pulling them closer to the center!

6. Live Parameter Sliders Guide

Manipulate the real-time physics sliders in the workbench to witness physical wave optics in action:

• Wavelength $\lambda$ (400–750 nm): Dynamically updates the rendered spectral color (violet $\to$ cyan $\to$ green $\to$ sodium yellow $\to$ red) and expands the fringe rings ($r \propto \sqrt{\lambda}$).
• Curvature Radius $R$ (0.50–3.00 m): Flatter lenses with larger $R$ expand the ring diameters ($r \propto \sqrt{R}$), making rings easier to measure in laboratory telescopes.
• Film Refractive Index $\mu$ (1.00–1.65): Increase from $1.00$ (Air) to $1.33$ (Water) or $1.55$ (Cedar oil) to observe the immediate contraction of ring diameters ($r \propto 1/\sqrt{\mu}$).
• Air Gap $t_0$ (0–300 nm): Simulates a dust speck lifting the lens. Watch how the central dark minimum continuously transitions into a bright maximum when $t_0 = \lambda/4$ ($\approx 147\text{ nm}$).