Ch 10 - WAVE OPTICS12th Physics

Corpuscular Model (Descartes/Newton): Predicted that light bending towards the normal (refracting into a denser medium) meant its speed would be greater in the second medium.

Wave Theory (Huygens): Predicted that if light bends towards the normal, its speed must be less in the second medium.

Conclusion: Foucault's experiment (1850) definitively proved the wave theory correct; light travels slower in water than in air.

Definition: A wavefront is the locus of all particles of a medium oscillating in the same phase at a given instant.

1. Spherical Wavefront: Emanates from a point source of light.
2. Cylindrical Wavefront: Emanates from a linear source (like a narrow slit).
3. Plane Wavefront: Formed by a small portion of a spherical or cylindrical wavefront originating from a very distant source (like the Sun).

Assumption 1: Every point on a given primary wavefront acts as a fresh source of secondary wavelets, which spread out in all directions with the speed of the wave in that medium.

Assumption 2: The new position of the wavefront at any later instant is the forward tangential envelope of these secondary wavelets at that instant.

Boundary Condition: The backward propagation of wavelets (the backwave) is strictly assumed to have an amplitude of zero.

Variables & Units:

• $i$ = Angle of incidence (rad or degrees)
• $r$ = Angle of refraction (rad or degrees)
• $v_1, v_2$ = Speed of the wave in medium 1 and 2 (m/s)
• $n_1, n_2$ = Absolute refractive indices of medium 1 and 2

Conditions/Notes:

• The derivation relies on the geometric fact that the time taken by secondary wavelets to travel from the primary wavefront to the refracting surface is constant.

Speed ($v$): Decreases ($v < c$) because the medium is optically denser.

Wavelength ($\lambda$): Decreases proportionally. The mathematical relation is $\frac{v_1}{v_2} = \frac{\lambda_1}{\lambda_2}$.

Frequency ($\nu$): Remains strictly constant. Frequency is an inherent characteristic of the oscillating source atoms, not the medium it travels through.

Phase Change: When a wave is reflected from a denser medium (e.g., traveling in air and bouncing off glass), it suffers a sudden, abrupt phase change of $\pi$ radians ($180^\circ$).

Path Difference Equivalent: This phase shift is mathematically equivalent to an additional optical path difference of $\lambda/2$.

Boundary Condition: A wave reflected from a rarer medium (e.g., traveling in glass and bouncing off air) suffers zero phase change.

It states that when two or more waves superimpose simultaneously at a point, the resultant displacement of the particle is exactly equal to the vector sum of the independent displacements produced by each individual wave.

Mathematically: $\vec{y} = \vec{y}_1 + \vec{y}_2 + \dots$

This principle forms the entire physical and mathematical foundation for both interference and diffraction phenomena.

Variables & Units:

• $\Delta \nu$ = Change in frequency (Hz)
• $\nu$ = Original frequency (Hz)
• $v_{\text{radial}}$ = Radial velocity of the source relative to the observer (m/s)
• $c$ = Speed of light in vacuum ($3 \times 10^8 \text{ m/s}$)

Conditions/Notes:

• Valid strictly for non-relativistic speeds ($v_{\text{radial}} \ll c$).
• Sign Convention: $v_{\text{radial}}$ is considered positive when the source moves away from the observer (yielding a negative frequency shift, i.e., Red Shift).

Coherent Sources: Sources that continuously emit light waves of the exact same frequency (or wavelength) and maintain a strictly constant phase difference (which can be zero) over time.

Incoherent Sources: Sources where the phase difference between the emitted waves changes rapidly and randomly over time (e.g., two separate, independent sodium lamps illuminating a wall).

Result: Coherent sources produce stable interference patterns, whereas incoherent sources simply add up their overall average intensities ($I = I_1 + I_2$) without forming sustained fringes.

Phase Difference ($\phi$): Must be an even integral multiple of $\pi$.
$\phi = 0, 2\pi, 4\pi, \dots = 2n\pi$

Path Difference ($\Delta x$): Must be a whole integral multiple of $\lambda$.
$\Delta x = 0, \lambda, 2\lambda, \dots = n\lambda$

Result: The amplitudes sum up in phase, producing maximum light intensity. If the two sources have equal intensity ($I_0$), the bright fringe has an intensity of $4I_0$.

Phase Difference ($\phi$): Must be an odd integral multiple of $\pi$.
$\phi = \pi, 3\pi, 5\pi, \dots = (2n-1)\pi$

Path Difference ($\Delta x$): Must be an odd multiple of $\lambda/2$.
$\Delta x = \frac{\lambda}{2}, \frac{3\lambda}{2}, \dots = \left(n - \frac{1}{2}\right)\lambda$

Result: The wave amplitudes exactly oppose and cancel each other. If the two sources have equal intensity ($I_1 = I_2$), the resultant intensity drops perfectly to zero.

Variables & Units:

• $I$ = Resultant Intensity ($W/m^2$)
• $I_1, I_2$ = Intensities of individual waves ($W/m^2$)
• $\phi$ = Phase difference between the two waves at the point of observation (rad)

Conditions/Notes:

• The term $2\sqrt{I_1 I_2} \cos \phi$ is formally known as the interference term.
• For identical sources ($I_1 = I_2 = I_0$), the formula simplifies beautifully to $I = 4I_0 \cos^2\left(\frac{\phi}{2}\right)$.

Definition: It is the phenomenon of light bending around the sharp corners of an obstacle or an aperture and physically spreading into the regions of the expected geometrical shadow.

Crucial Condition: Diffraction effects become distinctly noticeable and pronounced only when the size of the obstacle or aperture ($a$) is comparable to the wavelength ($\lambda$) of the incident light ($a \approx \lambda$).

Transverse Nature: Polarisation provides conclusive proof that light is a transverse wave (specifically, a transverse electromagnetic wave).

Mechanism: In transverse waves, oscillations occur strictly perpendicular to the direction of propagation, allowing them to be mathematically restricted to a single plane.

Comparison: Longitudinal waves (like sound waves in air) inherently cannot be polarised because their particle oscillations occur strictly parallel to the direction of wave propagation.

Unpolarised Light: The electric field vectors vibrate rapidly and randomly in all possible directions within the transverse plane perpendicular to the direction of wave propagation.

Plane Polarised Light (Linearly Polarised): The electric field vectors are strictly restricted to vibrate in one single direction (one plane) that remains constant over time.

Variables & Units:

• $\phi$ = Phase difference (rad)
• $\Delta x$ = Path difference (m)
• $\lambda$ = Wavelength of the wave (m)

Conditions/Notes:

• A spatial path difference of exactly one full wavelength ($\lambda$) conceptually and mathematically corresponds to a phase shift of one complete cyclic rotation ($2\pi$ radians).

Wavefront Splitting: He passed monochromatic light from a single primary pinhole/slit ($S$) to illuminate two closely spaced secondary slits ($S_1$ and $S_2$).

Phase Locking: Because the light at $S_1$ and $S_2$ originates from the exact same primary wavefront, any random, abrupt phase change in the original source $S$ manifests equally and simultaneously at $S_1$ and $S_2$.

Result: $S_1$ and $S_2$ act as perfectly coherent sources locked in phase, successfully generating stable, observable interference fringes on the screen.

Variables & Units:

• $\Delta x$ = Path difference (m)
• $x$ = Distance of the observation point P from the central maximum (m)
• $d$ = Separation distance between the two slits (m)
• $D$ = Distance from the plane of the slits to the viewing screen (m)

Conditions/Notes:

• The formula strictly relies on two paraxial approximations: $D \gg d$ and $x \ll D$, meaning the angle $\theta$ is extremely small so $\sin \theta \approx \tan \theta = \frac{x}{D}$.

Variables & Units:

• $x_n$ = Distance of the $n$-th bright fringe from the central maximum (m)
• $n$ = Order of the bright fringe ($0, \pm 1, \pm 2, \dots$)
• $\lambda$ = Wavelength of light (m)
• $D$ = Distance to screen (m)
• $d$ = Slit separation (m)

Conditions/Notes:

• Derived by equating the path difference $\frac{xd}{D}$ to the absolute condition for constructive interference ($n\lambda$).
• At $n=0$, $x_0 = 0$, forming the Central Bright Fringe exactly on the perpendicular bisector.

Variables & Units:

• $x_n$ = Distance of the $n$-th dark fringe from the central bright fringe (m)
• $n$ = Order of the dark fringe ($1, 2, 3, \dots$)
• $\lambda, D, d$ = Standard YDSE geometric parameters (m)

Conditions/Notes:

• Derived by equating the path difference $\frac{xd}{D}$ to the absolute condition for destructive interference $(n-\frac{1}{2})\lambda$.
• The very first dark fringe flanking the center occurs at $n=1$, meaning $x_1 = \frac{\lambda D}{2d}$.

Variables & Units:

• $\beta$ = Fringe width (m)
• $\lambda$ = Wavelength of the light (m)
• $D$ = Distance from the slit plane to the screen (m)
• $d$ = Separation distance between the two coherent slits (m)

Conditions/Notes:

• In Young's Double Slit Experiment, the widths of all bright fringes and all dark fringes are strictly equal.
• If the entire experimental setup is immersed in a liquid of refractive index $\mu$, the wavelength drops to $\lambda/\mu$, and the new fringe width shrinks identically: $\beta' = \frac{\beta}{\mu}$.

Variables & Units:

• Shift = Linear distance the entire pattern moves (m)
• $\mu$ = Refractive index of the inserted transparent plate
• $t$ = Thickness of the plate (m)
• $\beta$ = Original fringe width (m)

Conditions/Notes:

• The pattern unequivocally shifts towards the side of the slit covered by the glass plate.
• The actual fringe width ($\beta$) remains absolutely unchanged.
• The optical path length increases, introducing an additional path difference of $(\mu - 1)t$.

Variables & Units:

• $a$ = Linear width of the single slit (m)
• $\theta$ = Angle of diffraction to the minimum (rad or degrees)
• $n$ = Order of the minimum ($\pm 1, \pm 2, \pm 3, \dots$). Note: $n \neq 0$
• $\lambda$ = Wavelength of the incident light (m)

Conditions/Notes:

• Contrast this heavily with YDSE: $n\lambda$ is the path difference condition for maxima in interference, but for minima in single-slit diffraction.
• Mechanically, the slit is theoretically divided into an even number of equal zones ($2n$) that undergo complete destructive cancellation.

Variables & Units:

• $a$ = Linear width of the single slit (m)
• $\theta$ = Angle of diffraction to the maximum (rad or degrees)
• $n$ = Order of the secondary maximum ($1, 2, 3, \dots$)
• $\lambda$ = Wavelength of the incident light (m)

Conditions/Notes:

• These secondary maxima become progressively significantly weaker in intensity as $n$ increases because only a tiny unbound fraction of the slit (e.g., $1/3$, $1/5$) contributes constructively while the rest cancels out.

Variables & Units:

• $\beta_0$ = Linear width of the central maximum on the screen (m)
• $\lambda$ = Wavelength of the incident light (m)
• $D$ = Distance from the diffracting slit to the viewing screen (m)
• $a$ = Linear width of the slit (m)

Conditions/Notes:

• The central maximum is structurally exactly twice as wide as any of the secondary maxima (which have a width of $\lambda D/a$).
• Counter-intuitive note: If the physical slit is narrowed ($a$ decreases), the central bright maximum optically widens (spreads out more on the screen).

Variables & Units:

• $\Delta\theta$ = Angular resolution / limit of resolution (rad)
• $\lambda$ = Wavelength of incident light (m)
• $D$ = Diameter (aperture) of the objective lens (m)

Conditions/Notes:

• The formula determines the minimum angular separation required between two distant point sources (like stars) for them to be seen as distinct images.
• A larger diameter $D$ gives a smaller $\Delta\theta$, which means higher resolving power.

Variables & Units:

• $d_{\text{min}}$ = Minimum separation distance (m)
• $\lambda$ = Wavelength of light used (m)
• $n$ = Refractive index of the medium between the object and the objective lens
• $\beta$ = Semi-vertical angle subtended by the objective at the object (rad or degrees)

Conditions/Notes:

• The denominator term $n \sin \beta$ is known as the Numerical Aperture.
• Using an oil-immersion objective increases $n$, which decreases $d_{\text{min}}$ and strictly increases resolving power.

Variables & Units:

• $Z_F$ = Fresnel distance (m)
• $a$ = Size (width) of the aperture or slit (m)
• $\lambda$ = Wavelength of the incident light (m)

Conditions/Notes:

• This distance mathematically dictates the validity of ray optics.
• For distances $z \ll Z_F$, diffraction spreading is negligible, and light travels roughly in straight lines. For $z \gg Z_F$, wave optics must be strictly applied.

Variables & Units:

• $I$ = Transmitted intensity through the analyser ($W/m^2$)
• $I_0$ = Maximum intensity of the incident polarised light ($W/m^2$)
• $\theta$ = Angle between the pass-axes of the polariser and the analyser (rad or degrees)

Conditions/Notes:

• If $\theta = 0^\circ$ (parallel axes), $I = I_0$ (maximum transmission).
• If $\theta = 90^\circ$ (crossed polaroids), $I = 0$ (complete darkness).

Red Shift: Occurs when the source is moving away from the observer. The apparent wavelength increases, shifting the spectral lines towards the red end of the visible spectrum.

Blue Shift: Occurs when the source is moving towards the observer. The apparent wavelength decreases, shifting the spectral lines towards the violet/blue end of the spectrum.

Variables & Units:

• $i_B$ = Brewster's Angle / Polarising Angle (rad or degrees)
• $\mu$ (or $n_{21}$) = Refractive index of the denser reflecting medium relative to the rarer incident medium (dimensionless)

Conditions/Notes:

• Assumes unpolarised light is incident on the interface.
• If a polariser blocks the component perpendicular to the plane of incidence, and light strikes at $i_B$, there will be zero reflection (100% transmission).

Central Fringe: The central maximum remains perfectly white because all visible component colors have zero path difference there and constructively interfere simultaneously.

Coloured Fringes: On either side of the center, a few coloured fringes will uniquely emerge.

Order of Colours: The inner edge of the first fringe closest to the central white fringe will appear Red (because Blue destructively interferes first at a smaller angle, since $\lambda_{\text{blue}} < \lambda_{\text{red}}$).

After a few fringes, the wavelengths overlap completely out of phase and the pattern washes out into uniform white illumination.

Variables & Units:

• $I_{\text{max}}$ = Maximum intensity at constructive interference ($W/m^2$)
• $I_{\text{min}}$ = Minimum intensity at destructive interference ($W/m^2$)
• $I_1, I_2$ = Intensities of individual coherent waves
• $a_1, a_2$ = Amplitudes of individual coherent waves

Conditions/Notes:

• The Average Intensity over the entire pattern is given by $I_{\text{av}} = \frac{I_{\text{max}} + I_{\text{min}}}{2} = I_1 + I_2 = a_1^2 + a_2^2$.
• A highly tested concept for calculating amplitude ratios given fringe visibility.

Interference (e.g., YDSE): Results from the superposition of waves coming from two distinctly separate coherent sources. The fringes typically have perfectly equal intensity and equal width.

Diffraction: Results from the superposition of a continuous family of secondary wavelets originating from different parts of the exact same wavefront (like across a single slit). The central maximum is twice as wide and far brighter than all secondary maxima, whose intensities fall off rapidly.

Transmitted Intensity: It becomes exactly half of the initial incident intensity ($I = I_0 / 2$).

Reason: A polaroid only allows the component of the electric field parallel to its unique pass-axis to transmit. Since unpolarised light oscillates symmetrically and randomly in all directions, averaging the square of the cosine over all angles yields exactly $1/2$. Thus, exactly 50% of the wave's energy is absorbed.

Variables & Units:

• $n_1, n_2$ = Order of the bright fringes for light 1 and light 2 (integers)
• $\lambda_1, \lambda_2$ = Wavelengths of the two monochromatic light beams (m)

Conditions/Notes:

• Derived from equating the linear positions: $x = \frac{n_1 \lambda_1 D}{d} = \frac{n_2 \lambda_2 D}{d}$.
• This equation is fundamentally used to find the lowest integer combination of $n_1$ and $n_2$ to locate the nearest point from the central maximum where overlapping occurs.

Mechanism: Unpolarised incident sunlight causes the electrons in atmospheric gas molecules to oscillate in both transverse directions (in the plane perpendicular to the sun's rays).

The 90-Degree Effect: Accelerating charges do not radiate energy along their own axis of oscillation. An observer looking at exactly $90^\circ$ to the incident sunlight only receives radiation from electrons oscillating perpendicular to the observer's line of sight.

Result: The radiation component parallel to the observer's line of sight contributes zero intensity. Thus, the scattered light viewed at $90^\circ$ is completely linearly polarised.

Variables & Units:

• $n$ = Number of fringes shifted (dimensionless)
• $\mu$ = Refractive index of the transparent slab
• $t$ = Thickness of the slab (m)
• $\lambda$ = Wavelength of the incident light in a vacuum/air (m)

Conditions/Notes:

• The total extra optical path difference introduced is $\Delta x = (\mu - 1)t$.
• Equating this to the condition for $n$ bright fringes ($n\lambda$) yields the formula.

Condition: The reflected ray and the refracted ray must be exactly perpendicular ($90^\circ$) to each other.

Outcome: At this specific angle of incidence (Brewster's angle), the reflected light is 100% plane-polarised, with its electric field vector vibrating strictly perpendicular to the plane of incidence.

Note: The refracted (transmitted) ray remains only partially polarised.

Postulate: For artificially produced metamaterials where relative permittivity ($\epsilon_r < 0$) and relative permeability ($\mu_r < 0$) are both negative, the refractive index is strictly defined as $n = -\sqrt{\mu_r \epsilon_r}$.

Optical Consequence: When light enters such a medium from air, the phase travels away from the direction of propagation. A ray entering from the 1st quadrant will anomalously refract into the 3rd quadrant rather than the usual 4th quadrant, although Snell's Law mathematically still holds.

Variables & Units:

• RP = Resolving Power (rad$^{-1}$)
• $D$ = Diameter (aperture) of the objective lens or mirror (m)
• $\lambda$ = Wavelength of the incident light (m)

Conditions/Notes:

• Resolving Power directly indicates a telescope's ability to clearly separate two closely spaced celestial bodies.
• To maximize RP, telescopes are built with massive objective diameters ($D$).

Variables & Units:

• RP = Resolving Power (m$^{-1}$)
• $n$ = Refractive index of the medium between the objective lens and object
• $\beta$ = Semi-vertical angle subtended by the objective at the object
• $\lambda$ = Wavelength of light used (m)

Conditions/Notes:

• The denominator term $d_{\text{min}} = \frac{1}{RP}$ is the actual spatial resolution limit.
• Using an oil-immersion objective increases $n$, thereby significantly boosting the Resolving Power.

Variables & Units:

• $P$ = Percentage of polarisation (%)
• $I_{\text{max}}$ = Maximum transmitted intensity observed upon rotating the analyser ($W/m^2$)
• $I_{\text{min}}$ = Minimum transmitted intensity observed upon rotating the analyser ($W/m^2$)

Conditions/Notes:

• If $I_{\text{min}} = 0$, the light is 100% plane-polarised.
• If $I_{\text{max}} = I_{\text{min}}$, the percentage is 0%, indicating completely unpolarised light.