Wave Optics and Interference

Principle of Superposition and Concept of Interference

• Principle of Superposition: When two or more light waves propagate through the same medium simultaneously, the resultant displacement at any point is equal to the algebraic sum of the individual displacements produced by each wave:   $y = y_1 \pm y_2$   where $y$ is the resultant displacement of the waves, and $y_1$, $y_2$ are the individual displacements of the two interfering waves.

  • When the two waves propagate in the same direction, a positive sign is used: $y = y_1 + y_2$.

  • When the two waves propagate in opposite directions, a negative sign is used: $y = y_1 - y_2$.

• Definition of Interference: The modification in the distribution of light intensity in the region of superposition of two or more light waves is defined as Interference.

• Classification of Interference:

  • Constructive Interference:

  • Occurs when two or more light waves propagated simultaneously in the same medium meet in phase at the superposition region.

  • The resultant amplitude is equal to the sum of the individual amplitudes of the waves:       $a = a_1 + a_2 \quad \text{or} \quad Y = Y_1 + Y_2$

  • Path Difference Requirement: The optical path difference between the two light waves must be an integral multiple of the wavelength of light used:       $\text{Path Difference} = n\lambda \quad (n = 0, 1, 2, 3, \dots)$

  • This constructive overlap produces maximum light intensity, forming bright fringes.

  • Destructive Interference:

  • Occurs when light waves propagate in the same medium and meet in an out-of-phase relationship at the superposition region.

  • The resultant amplitude is equal to the difference of individual wave amplitudes:       $a = a_1 - a_2 \quad \text{or} \quad Y = Y_1 - Y_2$

  • Path Difference Requirement: The optical path difference must be equal to a half-integral multiple of the wavelength of light used:       $\text{Path Difference} = (2n + 1)\frac{\lambda}{2} \quad (n = 0, 1, 2, 3, \dots)$

  • This destructive overlap produces minimum or zero light intensity, forming dark fringes.

Coherence and Conditions for Sustained Interference

• Definition of Coherence: Light waves having the same wavelength, same frequency, same amplitude, and maintaining a constant phase difference over space and time are termed coherent waves, and their sources are called coherent sources.

• Coherent Length: Coherent length is defined as the propagation distance over which a coherent light wave maintains a specified degree of phase coherence.

• Types of Coherence:

  • Temporal Coherence (or Longitudinal Coherence):

  • Defined as the ability to predict the exact phase relation at a point on a wave with respect to another point along the direction of propagation on the same wave.

  • Spatial Coherence (or Transverse Coherence):

  • Defined as the ability to predict the exact phase relation at a point on one wave with respect to another point across a second wave orthogonal to the propagation direction.

• Essential Conditions for Sustained Interference Pattern Formation:

  1. Coherent Light Waves: The light waves emitting from the two sources must be coherent.

  2. Continuous Wavelength & Frequency: The two sources must emit continuous light waves of identical wavelength and frequency.

  3. Small Source Separation: The physical separation between the two light sources must be extremely small.

  4. Large Screen Distance: The distance between the light sources and the observation screen must be large relative to source separation.

  5. Dark Background: To clearly view high-contrast interference fringes, the background environment should be completely dark.

  6. Equal Amplitudes: The amplitudes of the two light waves should be equal or nearly equal to obtain complete darkness at minima.

  7. Narrow Sources: The light sources must be narrow slits.

  8. Monochromatic Light: The sources must emit monochromatic light (single wavelength $\lambda$).

Interference in Thin Films by Reflection

• Physical System: Consider a thin film of uniform thickness $t$ and refractive index $\mu$ surrounded by air (refractive index $\mu_{\text{air}} = 1$).

• Ray Tracing & Superposition:

  • A monochromatic ray of light $AB$ is incident at an angle $i$ on the upper boundary surface of the thin film.

  • At point $B$, part of the light is reflected along path $BE$ as Ray 1.

  • The remaining light refracts into the film along path $BC$ at an angle of refraction $r$.

  • At point $C$ on the lower boundary surface, the ray reflects internally along path $CD$.

  • At point $D$ on the upper surface, it refracts back into air along path $DF$ as Ray 2.

  • Reflected Ray 1 and Ray 2 superimpose to produce an interference pattern whose intensity distribution depends on their optical path difference.

• Derivation of Path Difference:

  • Construct a perpendicular $DG$ from point $D$ onto reflected Ray 1 ($BE$).

  • The path difference up to points $G$ and $D$ is given by:     $\text{Path Difference} = \text{Refractive index of thin film} \times (BC + CD) - \text{Refractive index of air} \times (BG)$     $\text{Path Difference} = \mu(BC + CD) - (1)(BG) \quad \text{--- (Equation 1)}$

  • Calculation of $(BC + CD)$:

  • From right-angled triangle $\triangle BCH$ (where $HC = t$):       $\cos(r) = \frac{HC}{BC} \implies BC = \frac{HC}{\cos(r)} = \frac{t}{\cos(r)}$

  • From right-angled triangle $\triangle DCH$ (where $HC = t$):       $\cos(r) = \frac{HC}{CD} \implies CD = \frac{HC}{\cos(r)} = \frac{t}{\cos(r)}$

  • Adding both segments:       $BC + CD = \frac{t}{\cos(r)} + \frac{t}{\cos(r)} = \frac{2t}{\cos(r)} \quad \text{--- (Equation 2)}$

  • Calculation of $BG$:

  • From $\triangle BHC$:       $\tan(r) = \frac{BH}{CH} \implies BH = CH\tan(r) = t\tan(r)$

  • From $\triangle DHC$:       $\tan(r) = \frac{HD}{CH} \implies HD = CH\tan(r) = t\tan(r)$

  • Total distance $BD$:       $BD = BH + HD = t\tan(r) + t\tan(r) = 2t\tan(r) \quad \text{--- (Equation 3)}$

  • From $\triangle BGD$:       $\sin(i) = \frac{BG}{BD} \implies BG = BD\sin(i) = 2t\tan(r)\sin(i)$

  • Applying Snell's Law ($\mu = \frac{\sin(i)}{\sin(r)} \implies \sin(i) = \mu\sin(r)$):       $BG = 2t\tan(r)\mu\sin(r) = 2\mu t\tan(r)\sin(r) \quad \text{--- (Equation 4)}$

  • Geometric Path Difference Substitution:

  • Substituting Equations 2 and 4 into Equation 1:       $\text{Path Difference} = \mu\left(\frac{2t}{\cos(r)}\right) - 2\mu t\tan(r)\sin(r)$       $\text{Path Difference} = \frac{2\mu t}{\cos(r)} - 2\mu t\left(\frac{\sin(r)}{\cos(r)}\right)\sin(r)$       $\text{Path Difference} = \frac{2\mu t}{\cos(r)}\left(1 - \sin^2(r)\right)$

  • Using trigonometric identity $1 - \sin^2(r) = \cos^2(r)$, the geometric path difference becomes:       $\text{Path Difference} = \frac{2\mu t}{\cos(r)}\cos^2(r) = 2\mu t\cos(r)$

• Stokes' Phase Change Correction:

  • At point $B$, reflection occurs at the surface of an optically denser medium (film boundary).

  • According to Stokes' principle, reflection at an optically denser medium introduces an abrupt additional phase shift of $\pi$ radians, corresponding to an additional path difference of $\frac{\lambda}{2}$.

  • Total Optical Path Difference:     $\text{Total Path Difference} = 2\mu t\cos(r) + \frac{\lambda}{2}$

• Fringe Conditions:

  • Condition for Bright Fringe (Constructive Interference):

  • When total path difference equals an integral multiple of wavelength ($n\lambda$):       $2\mu t\cos(r) + \frac{\lambda}{2} = n\lambda$       $2\mu t\cos(r) = n\lambda - \frac{\lambda}{2}$       $2\mu t\cos(r) = (2n - 1)\frac{\lambda}{2} \quad (n = 1, 2, 3, \dots)$

  • Condition for Dark Fringe (Destructive Interference):

  • When total path difference equals a half-integral multiple of wavelength ($(2n + 1)\frac{\lambda}{2}$):       $2\mu t\cos(r) + \frac{\lambda}{2} = (2n + 1)\frac{\lambda}{2}$       $2\mu t\cos(r) + \frac{\lambda}{2} = n\lambda + \frac{\lambda}{2}$       $2\mu t\cos(r) = n\lambda \quad (n = 0, 1, 2, 3, \dots)$

Formation of Colours in Thin Films by Reflection

• Phenomenon: When a soap bubble or thin oil layer floating on water is illuminated by white solar radiation, brilliant color bands appear.

• Mechanism:

  • Solar light consists of a continuous spectrum of wavelengths ($\lambda$).

  • The constructive condition $2\mu t\cos(r) = (2n - 1)\frac{\lambda}{2}$ depends on refractive index $\mu$, film thickness $t$, angle of refraction $r$, and wavelength $\lambda$.

  • For a particular thickness $t$ and viewing angle $r$, only specific colors (wavelengths) satisfy the constructive interference condition and are strongly reflected toward the observer.

  • Other wavelengths fulfill the destructive condition $2\mu t\cos(r) = n\lambda$ and vanish from reflected light.

  • Because film thickness $t$ varies continuously across a soap bubble or floating oil film, different colors meet constructive conditions at different locations, displaying vibrant color patterns.

Newton's Rings: Experimental Setup and Theory

• Definition: Newton's rings are an example of interference produced by a non-uniform air film of radially increasing thickness. The pattern was discovered by Sir Isaac Newton and consists of concentric alternate bright and dark circular rings.

• Experimental Arrangement:

• Apparatus Elements:

  • A monochromatic light source.

  • A plano-convex lens $L$ resting on a flat plane glass plate $P$ (contact point $O$).

  • A glass plate $G$ inclined at an angle of $45^\circ$ relative to the incident light path.

  • A travelling microscope positioned vertically above plate $G$ for viewing rings.

• Operation:

  • Monochromatic light strikes glass plate $G$ at $45^\circ$ and reflects vertically downward, falling at normal incidence ($i = 0^\circ$) onto the wedge-shaped air film formed between lens $L$ and plate $P$.

• Ray Formation Mechanism:

  • Light ray $AB$ falls normally on glass plate $G$ and is reflected vertically downward onto the air film.

  • Ray 1 reflects from the upper curved air film surface (lower surface of lens $L$ at point $B$), acquiring a phase shift of $\frac{\lambda}{2}$, and emerges through the lens.

  • Ray 2 transmits through the air film, reflects from the top surface of glass plate $P$ at point $C$ ( acquiring an additional phase shift of $\frac{\lambda}{2}$), re-enters lens $L$, and emerges as Ray 2.

  • Rays 1 and 2 superimpose to form circular interference fringes.

• Effective Path Difference:

  • General reflection path difference: $2\mu t\cos(r) + \frac{\lambda}{2}$.

  • For normal incidence $r = 0^\circ \implies \cos(0^\circ) = 1$, and for air medium $\mu = 1$:       $\text{Path Difference} = 2(1)t\cos(0^\circ) + \frac{\lambda}{2} = 2t + \frac{\lambda}{2}$

  • Nature of the Central Spot:

• At the central point of contact $O$, film thickness t = 0$.\n - \text{Path Difference} = 2(0) + \frac{\lambda}{2} = \frac{\lambda}{2}.\n - A path difference of \frac{\lambda}{2}$corresponds to a phase shift of$\pi, causing the incident and reflected light to meet completely out of phase.\n - Destructive interference occurs at point O, rendering the **central spot completely dark**.\n\n- **Ring Conditions**:\n - **Bright Rings**: 2t + \frac{\lambda}{2} = n\lambda \implies 2t = (2n - 1)\frac{\lambda}{2} \quad (n = 1, 2, 3, \dots)\n - **Dark Rings**: 2t + \frac{\lambda}{2} = (2n + 1)\frac{\lambda}{2} \implies 2t = n\lambda \quad (n = 0, 1, 2, 3, \dots)\n\n# Mathematical Expression for Diameters of Newton's Rings\n\n- **Geometric Derivation**:\n\n\n\n - Let R$be the radius of curvature of the convex lens surface,$r$be the radius of a Newton's ring, and$t$be the film thickness at radius$r.\n - From circle geometry (theorem of intersecting chords):\n    NA \times NB = NO \times ND\n    r \times r = t \times (2R - t)\n    r^2 = 2Rt - t^2\n - Since thickness t$is extremely small relative to radius of curvature$R$,$t^2$is negligible ($t^2 \approx 0):\n    r^2 = 2Rt \implies t = \frac{r^2}{2R}\n\n- **Diameter of Bright Rings**:\n - Substituting t = \frac{r^2}{2R}$into the bright ring condition$2t = (2n - 1)\frac{\lambda}{2}:\n    2\left(\frac{r^2}{2R}\right) = (2n - 1)\frac{\lambda}{2} \implies \frac{r^2}{R} = \frac{(2n - 1)\lambda}{2} \implies r^2 = \frac{(2n - 1)\lambda R}{2}\n - Substituting r = \frac{D}{2}$(where$D is the ring diameter):\n    \left(\frac{D}{2}\right)^2 = \frac{(2n - 1)\lambda R}{2} \implies \frac{D^2}{4} = \frac{(2n - 1)\lambda R}{2}\n    D^2 = (2n - 1)(2\lambda R)\n    D = \sqrt{2n - 1}\sqrt{2\lambda R}\n    D \propto \sqrt{2n - 1}\n - **Conclusion**: The diameters of bright Newton's rings are directly proportional to the square root of odd natural numbers (\sqrt{1}, \sqrt{3}, \sqrt{5}, \dots).\n\n- **Diameter of Dark Rings**:\n - Substituting t = \frac{r^2}{2R}$into the dark ring condition$2t = n\lambda:\n    2\left(\frac{r^2}{2R}\right) = n\lambda \implies \frac{r^2}{R} = n\lambda \implies r^2 = n\lambda R\n - Substituting r = \frac{D}{2}:\n    \left(\frac{D}{2}\right)^2 = n\lambda R \implies \frac{D^2}{4} = n\lambda R\n    D^2 = 4n\lambda R\n    D = \sqrt{4n\lambda R} = 2\sqrt{n}\sqrt{\lambda R}\n    D \propto \sqrt{n}\n - **Conclusion**: The diameters of dark Newton's rings are directly proportional to the square root of natural numbers (\sqrt{1}, \sqrt{2}, \sqrt{3}, \dots).\n - **Fringe Spacing Trend**: As ring order n increases, the difference between consecutive square roots decreases, causing rings to get progressively closer together and reducing fringe width outward.\n\n# Experimental Determination of Wavelength and Refractive Index\n\n- **Determination of Wavelength (\lambda) of a Light Source**:\n - Measure diameters D_m$and$D_n$of the$m^{\text{th}}$and$n^{\text{th}}$dark rings ($n > m) using a travelling microscope.\n - Squared diameters for n^{\text{th}}$and$m^{\text{th}} dark rings:\n    D_n^2 = 4n\lambda R\n    D_m^2 = 4m\lambda R\n - Subtracting equations:\n    D_n^2 - D_m^2 = 4n\lambda R - 4m\lambda R = 4\lambda R(n - m)\n - **Wavelength Formula**:\n    \lambda = \frac{D_n^2 - D_m^2}{4R(n - m)}\n - **Radius of Curvature Formula**:\n    R = \frac{D_n^2 - D_m^2}{4\lambda(n - m)}\n\n- **Determination of Refractive Index (\mu) of a Liquid**:\n - First perform the experiment with an air film (\mu = 1) between the lens and glass plate.\n - Measure diameters D_m$and$D_n$of$m^{\text{th}}$and$n^{\text{th}} dark rings:\n    D_n^2 - D_m^2 = 4\lambda R(n - m) \quad \text{--- (Equation 1)}\n - Introduce the liquid film of refractive index \mu between the lens and glass plate.\n - Measure new diameters D_m'$and$D_n'$for the same$m^{\text{th}}$and$n^{\text{th}} dark rings:\n    (D_n')^2 - (D_m')^2 = \frac{4\lambda R(n - m)}{\mu} \quad \text{--- (Equation 2)}\n - Divide Equation 1 by Equation 2:\n    \frac{D_n^2 - D_m^2}{(D_n')^2 - (D_m')^2} = \frac{4\lambda R(n - m)}{\frac{4\lambda R(n - m)}{\mu}} = \mu\n - **Refractive Index Formula**:\n    \mu = \frac{D_n^2 - D_m^2}{(D_n')^2 - (D_m')^2}\n\n# Fundamental Conceptual Questions and Answers\n\n- **Why Two Independent Light Sources Cannot Produce Interference Fringes**:\n - Independent monochromatic sources emit light via spontaneous emission where individual atoms radiate wave packets with independent, randomly fluctuating initial phases.\n - Consequently, they cannot maintain a constant phase difference (\Delta\phi) over time, violating coherence requirements and preventing stable interference patterns.\n\n- **Why Distinct Rays from a Single Monochromatic Lamp Cannot Interfere**:\n - Light emitted spontaneously from spatially distinct regions of a single lamp exhibits independent random phase shifts, preventing coherence across distinct rays unless split systematically from the same wavefront segment.\n\n- **Method to Produce Coherent Light Waves**:\n - A single wavefront from a monochromatic source is split into two secondary wavefronts using two narrow slits (as in Young's double slit configuration) or amplitude division (thin film reflections). Because both secondary sources originate from the same primary wavefront, phase fluctuations occur synchronously, maintaining a constant phase difference.\n\n- **Verification of Conservation of Energy in Destructive Interference**:\n - Destructive interference does not violate the law of conservation of energy.\n - Minimum intensity at dark fringe: I_{\min} \propto (a - a)^2 = 0\n - Maximum intensity at bright fringe: I_{\max} \propto (a + a)^2 = 4a^2\n - Average intensity across a region containing one maximum and one minimum:\n    I_{\text{avg}} = \frac{I_{\max} + I_{\min}}{2} = \frac{4a^2 + 0}{2} = 2a^2\n - Total un-interfered light intensity from two independent waves of amplitude a:\n    I_{\text{total}} \propto a^2 + a^2 = 2a^2\n - Since average intensity 2a^2$equals un-interfered combined intensity$2a^2, light energy is not destroyed; energy is merely redistributed from destructive regions to constructive regions.\n\n- **Summary of Applications of Interference**:\n - Testing optical flatness of surfaces.\n - Testing optical quality and surface curvature of lenses.\n - Determining thickness of thin transparent films.\n - Constructing optical interference filters to isolate specific spectral radiation.\n - Precise determination of wavelength (\lambda) of light sources.\n - Accurate measurement of refractive index (\mu$$) of liquids.