Wave Optics - Class 12

A light source emitting a continuous spectrum of photon energies from $E_{min} = 1.50 \text{ eV}$ to $E_{max} = 4.00 \text{ eV}$ illuminates a Young's Double Slit setup with a slit separation of $d = 1.00 \text{ mm}$. A screen is placed at a distance $D = 2.00 \text{ m}$ from the slits. The screen is coated with a photoelectric material having a work function $\Phi = 2.48 \text{ eV}$, and the interference pattern is recorded by mapping the local photoelectric current density on the screen. Determine the minimum distance $y > 0$ from the central maximum where a bright fringe of the shortest detected wavelength exactly coincides with a dark fringe of the longest detected wavelength. (Take $hc = 1240 \text{ eV}\cdot\text{nm}$)

In a Fresnel's biprism experiment performed in air, a monochromatic point source of light of wavelength $\lambda$ is placed at a distance $a$ from a biprism. The biprism has a refractive index $n_g = 1.5$ and a small refracting angle $\alpha$. The interference fringes are observed on a screen placed at a distance $L$ from the source. The observed fringe width is $\beta_0 = 0.30 \text{ mm}$. The entire apparatus is now completely immersed in a transparent liquid of refractive index $n_l = \frac{4}{3}$. Assuming the small angle approximation holds, what is the new fringe width $\beta'$ observed on the screen?

In a modified Young's Double Slit Experiment, the entire setup is immersed in a lossy liquid medium. The amplitude of a light wave traveling a distance $x$ in this medium decays as $A(x) = A_0 e^{-\alpha x}$, where $\alpha$ is a positive absorption constant. The slits $S_1$ and $S_2$ are separated by a distance $d$ and are illuminated by coherent light such that the initial amplitudes at the slits are both $A_0$. The screen is placed at a distance $D$ ($D \gg d$). If the wavelength of the light in the medium is $\lambda$, which of the following expressions correctly represents the visibility $V$ of the $n^{\text{th}}$ bright fringe on the screen? \n\n(Assume $y \ll D$ where $y$ is the distance from the central maximum, and define local visibility as $V = \frac{I_{\text{max}} - I_{\text{min}}}{I_{\text{max}} + I_{\text{min}}}$ where $I_{\text{max}}$ and $I_{\text{min}}$ are the envelope intensities near the $n^{\text{th}}$ fringe.)

A beam of non-relativistic electrons, each of mass $m$ and initial kinetic energy $E$, is incident on a double-slit apparatus with slit separation $d$. A screen is placed at a distance $D$ from the slits ($D \gg d$). In the region between the slits and the screen, a uniform electric field is applied perpendicular to the plane of the slits. This field accelerates the electrons such that their kinetic energy upon reaching the screen is $4E$. If $\lambda_0$ is the initial de Broglie wavelength of the electrons before entering the slits, what is the fringe width $\beta$ of the interference pattern formed on the screen? (Assume the small-angle approximation holds and ignore gravitational effects).

A horizontal wind tunnel of length $L$ is designed such that the flow velocity of an ideal gas along its central axis $x$ (from $x=0$ to $x=L$) is given by $v(x) = v_0 \sqrt{1 + \frac{x}{L}}$. The gas has molar mass $M$ and is maintained at a constant temperature $T$. At $x=0$, the gas density is $\rho_0$. Assume the flow is steady and the density variations are small enough that the fluid can be treated as approximately incompressible when applying Bernoulli's equation. The refractive index of the gas depends on its local density $\rho$ according to the Gladstone-Dale relation $n(x) = 1 + K\rho(x)$, where $K$ is a positive constant. A laser beam of vacuum wavelength $\lambda$ travels along the central axis from $x=0$ to $x=L$. What is the magnitude of the phase shift $\Delta \phi$ of this laser beam relative to a reference beam traveling the same distance $L$ through a uniform stationary gas of constant density $\rho_0$? (Here $R$ is the universal gas constant).

In a Young's Double Slit Experiment, light of wavelength $\lambda$ illuminates two slits separated by an initial distance $d_0$. The interference pattern is recorded on a photographic plate placed at a distance $D$ ($D \gg d_0$). During the exposure time $T$, the plane containing the slits is uniformly heated such that the slit separation increases linearly with time, reaching a final separation of $d_0 + \Delta d$. The plate records the time-averaged intensity $\langle I(y) \rangle = \frac{1}{T} \int_0^T I(y,t) dt$, where $y$ is the distance from the central maximum. What is the minimum non-zero distance $y$ where the fringe contrast completely vanishes?

In a Young's Double Slit Experiment, two slits $S_1$ and $S_2$ are illuminated by monochromatic light of wavelength $\lambda$. The intensity of light from each slit on the screen is $I_0$. A transparent dielectric slab of refractive index $\mu$ is placed in front of slit $S_1$. The thickness of the slab varies harmonically with time as $d(t) = d_0 + A \sin(\omega t)$, where $\omega$ is much greater than the reciprocal of the detector's response time. Given that $d_0 = \frac{\lambda}{3(\mu-1)}$ and $A = \frac{\lambda \beta}{2\pi(\mu-1)}$ (where $\beta$ is a positive dimensionless constant). The detector measures the time-averaged intensity $\langle I \rangle$ at the central point $O$ on the screen (where the geometric path difference is zero). You may use the integral identities: $\frac{1}{2\pi} \int_0^{2\pi} \cos(\beta \sin \theta) d\theta = J_0(\beta)$ and $\frac{1}{2\pi} \int_0^{2\pi} \sin(\beta \sin \theta) d\theta = 0$. Which of the following expressions correctly represents the measured intensity $\langle I \rangle$ at point $O$?

A standard Young's Double Slit Experiment setup has two slits $S_1$ and $S_2$ separated by a distance $d = 0.5 \text{ mm}$, placed symmetrically on the y-axis at $x=0$. The slits are illuminated by coherent light of wavelength $\lambda = 600 \text{ nm}$. The observation screen is replaced by a uniform circular disc of mass $M = 2.0 \text{ kg}$ and radius $R = 0.3 \text{ m}$, lying in the plane $x = D = 0.4 \text{ m}$. The disc is mounted on a frictionless axle along the x-axis passing through its center $(D, 0, 0)$, and is initially at rest. A small sticky particle of mass $m = 0.5 \text{ kg}$ moving with velocity $\vec{v} = v_0 \hat{k}$ where $v_0 = 15 \text{ m/s}$, strikes the edge of the disc at the point $(D, y_0, z_0)$ where $y_0 = 0.18 \text{ m}$ and $z_0 = 0.24 \text{ m}$, and embeds itself. Due to the rotation of the disc, the effective path length from $S_1$ and $S_2$ to any point on the disc changes continuously, causing the observed intensity to oscillate with a beat frequency $\Delta f$. Assuming $d \ll D$ and $d \ll R$, find the maximum beat frequency $\Delta f_{\text{max}}$ observed anywhere on the disc after the collision.

A point source located at the origin emits continuous transverse waves. The waves propagate spherically outward through a non-ideal, isotropic medium. Due to the interaction of secondary wavelets with the medium, energy is continuously dissipated. The rate of total power loss with respect to the radial distance $r$ is given by $\frac{dP}{dr} = - 8\pi \alpha r I(r)$ where $I(r)$ is the local wave intensity and $\alpha$ is a dimensionless positive constant. If the amplitude of the wave at a distance $r_0$ is $A_0$, what is the amplitude of the wave at a distance $2r_0$?

In a modified Young's Double Slit Experiment, the space between the slit plane (at $x = 0$) and the screen (at $x = D$) is completely filled with a non-uniform transparent medium. The refractive index of this medium varies spatially as $n(y) = n_0 + k y$, where $y$ is the transverse coordinate parallel to the slit plane, $n_0$ is a dimensionless constant, and $k$ is a positive constant in $\text{m}^{-1}$. The two slits $S_1$ and $S_2$ are located at $y = d/2$ and $y = -d/2$ respectively. Assuming $d \ll D$ and considering only paraxial rays ($y \ll D$) such that the geometric paths can be approximated as straight lines, find the $y$-coordinate of the central bright fringe on the screen.

A thin transparent film of thickness $t$ and refractive index $n_2 = \sqrt{2}$ is coated on a thick glass substrate of refractive index $n_3 = 2$. The film is in contact with air ($n_1 = 1$). A plane light wave of wavelength $\lambda$ (in vacuum) is incident on the film from air at an angle of incidence $\theta_i = 60^\circ$. The incident light is p-polarized (electric field in the plane of incidence).

The amplitude reflection coefficient for p-polarized light moving from medium $n_a$ to $n_b$ is given by $r_p = \frac{n_b \cos \theta_a - n_a \cos \theta_b}{n_b \cos \theta_a + n_a \cos \theta_b}$, where $\theta_a$ and $\theta_b$ are the angles of incidence and refraction, respectively. A positive $r_p$ corresponds to a phase shift of $\pi$, while a negative $r_p$ corresponds to a phase shift of $0$.

Find the minimum non-zero thickness $t$ of the film for which the reflected light undergoes perfectly destructive interference.

Two identical coherent sources $S_1$ and $S_2$ are placed at the ends of a narrow 1D waveguide of length $d = 10\text{ m}$. They emit waves in phase with the same initial amplitude. The waveguide is filled with a dissipative medium where the amplitude of a wave decays exponentially with distance, halving every $5\text{ m}$. A detector moves along the waveguide to observe the interference pattern. Assuming the wavelength $\lambda$ is very small compared to the attenuation length, what is the fringe visibility $V = \frac{I_{\text{max}} - I_{\text{min}}}{I_{\text{max}} + I_{\text{min}}}$ of the local interference pattern at a distance of $x = 2.5\text{ m}$ from $S_1$?

A plane monochromatic wave of wavelength $\lambda$ is incident normally on a rectangular slit of width $a$. A transparent glass plate of uniform thickness $t$ is placed immediately behind the slit, covering it completely. The refractive index of the glass varies with temperature $T$ as $\mu(T) = \mu_0 + \gamma T$, where $\mu_0$ and $\gamma$ are positive constants. A steady temperature gradient is established along the width of the slit such that the temperature varies linearly from $T_0$ at the lower edge to $T_0 + \Delta T$ at the upper edge. Assuming thermal expansion of the glass and the slit is negligible, what is the minimum positive temperature difference $\Delta T$ required so that the intensity of light at the geometric center of the diffraction pattern (i.e., at an angle $\theta = 0^\circ$) becomes zero?

Two coherent, plane electromagnetic waves of equal amplitude propagate in a homogeneous, isotropic, linear dielectric medium of relative permittivity $\kappa = 4$ and relative permeability $\mu_r = 1$. Wave 1 propagates along the $+x$-axis and is linearly polarized with its electric field vector $\vec{E}_1$ oscillating parallel to the $+z$-axis. Wave 2 propagates along the $+y$-axis and is linearly polarized with its electric field vector $\vec{E}_2$ oscillating parallel to the $+z$-axis. The electric field amplitude of each wave in the medium is $E_0$. Consider a point in the medium where the electric fields of the two waves interfere constructively to yield a local maximum amplitude. At this point, what is the ratio of the time-averaged polarization energy density $\langle u_P \rangle$ to the time-averaged magnetic energy density $\langle u_B \rangle$?

(Note: The polarization energy density is defined as $u_P = \frac{1}{2} \vec{P} \cdot \vec{E}$, where $\vec{P}$ is the macroscopic polarization vector).

In a modified Young's Double Slit Experiment, monochromatic light of wavelength $\lambda$ illuminates two identical slits $S_1$ and $S_2$ separated by a distance $d$. The screen is placed at a distance $D \gg d$. Slit $S_1$ is fixed, while slit $S_2$ is made to oscillate along the axis perpendicular to the plane of the slits with a displacement $z(t) = A \sin(\omega t)$, where $A \ll \lambda$. The frequency of oscillation $f = \omega / 2\pi$ is much higher than the response rate of the detector on the screen, so the detector records the time-averaged intensity. If the intensity produced by a single slit on the screen is $I_0$, what is the fringe visibility $V = \frac{I_{\text{max}} - I_{\text{min}}}{I_{\text{max}} + I_{\text{min}}}$ of the observed interference pattern?

A free, excited atom of mass $m$ is initially at rest at a distance $L$ from a double-slit apparatus with slit separation $d$. It emits a single photon of wavelength $\lambda$ which passes through the slits and is detected on a screen at a distance $D$ from the slits. An experimenter measures the transverse recoil velocity $v_y$ (parallel to the slit separation) of the atom to an accuracy $\Delta v_y = \frac{h d}{2 m \lambda L}$ in an attempt to determine which slit the photon passed through ($h$ is Planck's constant). Assuming the atom's transverse position and momentum satisfy the minimum uncertainty condition $\Delta y \Delta p_y = \frac{h}{4\pi}$, what is the resulting minimum uncertainty in the position of the central interference maximum on the screen?

In a Young's Double Slit Experiment, the slits $S_1$ and $S_2$ are separated by a distance $d = 1 \text{ mm}$ and are illuminated by monochromatic light of wavelength $\lambda = 600 \text{ nm}$. A screen is placed at a distance $D = 2 \text{ m}$ from the slits. The entire setup is initially in a vacuum. A thin transparent plate of uniform thickness $t = 2 \text{ mm}$ is placed parallel to the plane of the slits, at an exact distance of $D/2 = 1 \text{ m}$ from the slits. The refractive index of this plate varies along the transverse $y$-direction (parallel to the slit separation) as $n(y) = 1 + \beta y^2$, where $\beta = 100 \text{ m}^{-2}$ and $y=0$ corresponds to the central axis passing through the midpoint of the slits. Assuming the deviation of light rays from straight lines is negligible, what is the new fringe width on the screen?

In a Young's Double Slit Experiment, the entire apparatus is kept in a vacuum. A transparent cylindrical tube of length $L$ is placed immediately in front of slit $S_1$. The tube contains an ideal gas initially at temperature $T_0$, and its initial refractive index is $n_0$. The gas is heated isobarically such that its temperature increases linearly with time as $T(t) = T_0 + \gamma t$, where $\gamma$ is a positive constant. The refractive index $n$ of the gas satisfies the Gladstone-Dale relation $(n - 1) \propto \rho$, where $\rho$ is the density of the gas. Ignore the thermal expansion of the tube. The slit separation is $d$, the screen distance is $D$, and the wavelength of the light in vacuum is $\lambda$. Which of the following expressions correctly gives the instantaneous fringe width $\beta(t)$ on the screen and the speed $v(t)$ of the central maximum?

Consider a flat circular disk of radius $R$ lying in the $x-y$ plane, centered at the origin. The entire disk acts as a coherent source of scalar waves, oscillating in phase at an angular frequency $\omega$. According to the Huygens-Fresnel principle, each infinitesimal area element $dA$ on the disk emits a spherical secondary wavelet of the form $d\psi = \frac{\psi_0}{r'} e^{i(kr' - \omega t)} dA$, where $r'$ is the distance from the element to the observation point, $k = \omega/c$, and $c$ is the speed of the wave in the medium. Ignore any obliquity factors. By integrating the contributions from the entire continuous distribution, determine the effective phase velocity $v_p$ of the resultant wave along the central $z$-axis as a function of the distance $z$ from the disk.

In a model bridging Huygens' principle and Maxwell's equations, a plane electromagnetic wave propagates in vacuum along the $+z$-axis. At $z=0$, the electric field is given by $\vec{E}(t) = E_0 \cos(\omega t) \hat{i}$. Consider a thin planar slice of the wavefront of thickness $\Delta z$ (where $\Delta z \ll c/\omega$) centered at $z=0$. This slice acts as a localized secondary source generating a magnetic field via its displacement current. What is the magnetic field contribution $\Delta \vec{B}(t)$ generated solely by the displacement current of this specific slice at $z = \Delta z/2$, and what is the phase difference $\Delta \phi = \phi_{\Delta B} - \phi_E$ between this magnetic field contribution and the local electric field $\vec{E}(t)$? (Assume $c$ is the speed of light in vacuum, and all quantities are in standard SI units).

A plane electromagnetic wavefront traveling in a vacuum grazes the apex of a perfectly absorbing, opaque obstacle. The cross-section of the obstacle in the $xy$-plane is given by the curve $y = \frac{2}{3\sqrt{L_0}} x^{3/2}$ for $x \ge 0$, where $L_0$ is a positive characteristic length. According to the Geometrical Theory of Diffraction, the grazing ray excites a creeping wave at the origin $(0,0)$ at time $t=0$. This creeping wave travels along the surface of the obstacle at the speed of light $c$, continuously shedding diffracted rays tangentially into the surrounding vacuum. At time $t = \frac{20 L_0}{3 c}$, what is the radius of curvature of the diffracted wavefront (in the $xy$-plane) at the position reached by the specific ray that was shed from the point $x = 3L_0$ on the obstacle?

Two plane electromagnetic waves, Wave 1 and Wave 2, propagate in vacuum along the $+z$-axis. Their electric field vectors are given by: $\vec{E}_1(z,t) = E_0 \cos(k_1 z - \omega_1 t) \hat{i}$ $\vec{E}_2(z,t) = E_0 \cos(k_2 z - \omega_2 t) \left( \cos\theta \hat{i} + \sin\theta \hat{j} \right)$ where $E_0$ is the electric field amplitude, $\theta = 60^\circ$, and $\omega_1, \omega_2$ are optical frequencies such that $\omega_1 - \omega_2 = \delta > 0$. The waves fall normally on a perfectly absorbing detector of area $A$ placed at $z = 0$. The detector has a response time much larger than $2\pi/\omega_1$ but much smaller than $2\pi/\delta$, so it measures the instantaneous intensity $I(t)$ averaged over the fast optical cycles. If $I_0 = \frac{1}{2}\epsilon_0 c E_0^2$ is the intensity of a single wave, what is the total electromagnetic energy absorbed by the detector during the time interval from $t = 0$ to $t = \frac{\pi}{2\delta}$?

A solid non-conducting sphere of radius $R$ has a spherically symmetric volume charge density $\rho(r) = \rho_0 \frac{r}{R}$, where $r$ is the radial distance from its center and $\rho_0$ is a positive constant. The solid sphere is surrounded by a concentric, grounded conducting spherical shell of inner radius $2R$ and outer radius $3R$. A point charge $q$ is placed at a distance $4R$ from the center of the system. What is the net electrostatic potential at the center of the solid sphere? (Assume the potential at infinity is zero, and $\epsilon_0$ is the permittivity of free space).

A stationary source at $x=0$ continuously emits a composite wave consisting of two harmonic components with angular frequencies $\omega_1 = \omega_0 + \Delta \omega$ and $\omega_2 = \omega_0 - \Delta \omega$, where $\Delta \omega \ll \omega_0$. The waves propagate in the $+x$ direction through a non-linear dispersive medium. The dispersion relation of the medium is given by $k(\omega) = \frac{\omega}{v_0} + \beta \omega^2$, where $v_0$ and $\beta$ are positive constants. At any given instant $t$, what is the spatial distance $\Delta x$ between two adjacent nodes of the beat envelope?

A monochromatic plane wave of wavelength $\lambda = 600 \text{ nm}$ (in vacuum) is incident on an opaque screen containing two identical circular pinholes. The plane of incidence contains the centers of the two pinholes, and the incident light rays make an angle $\theta_0$ with the normal to the screen, where $\sin \theta_0 = 0.75$. The diameter of each pinhole is $a = 1.22 \text{ } \mu\text{m}$ and the distance between their centers is $d = 4.00 \text{ } \mu\text{m}$. The entire setup is immersed in a transparent liquid of refractive index $\mu = 1.5$. The interference pattern is observed on a large semicircular screen placed in the forward direction in the plane of incidence, covering observation angles $-\pi/2 \le \theta \le \pi/2$ (where $\theta$ is measured from the normal). How many interference principal maxima of non-zero intensity are observed on the screen strictly within the central diffraction envelope (Airy disk) of the individual pinholes?

Two coherent monochromatic electromagnetic waves, Wave 1 and Wave 2, propagate in a vacuum along the $+z$ direction. Their electric field vectors are given by: ${\vec{E}_1 = E_0 \cos(kz - \omega t) \hat{i}}$ ${\vec{E}_2 = E_0 \cos\left(kz - \omega t + \frac{\pi}{3}\right) \hat{j}}$ where $E_0$ is a constant amplitude in $\text{V/m}$, $k$ is the wave number in $\text{rad/m}$, and $\omega$ is the angular frequency in $\text{rad/s}$. The time-averaged intensity of Wave 1 alone is $I_0$ (in $\text{W/m}^2$). The two waves are superposed, and the resultant wave is incident normally on an ideal linear polarizer. The transmission axis of the polarizer lies in the $xy$-plane and makes an angle of $45^\circ$ with the positive $x$-axis. What is the time-averaged intensity of the transmitted wave?

In a Young's double slit experiment, the distance between the slits is $d = 1.0 \text{ mm}$ and the screen is at a distance $D = 1.0 \text{ m}$ from the slits. A light beam containing two wavelengths $\lambda_1 = 400 \text{ nm}$ and $\lambda_2 = 600 \text{ nm}$ is incident normally on the slits. A thin transparent slab of thickness $t = 2.0 \text{ } \mu\text{m}$ is placed in front of the upper slit. The material of the slab is dispersive, with a refractive index $\mu_1 = 1.60$ for $\lambda_1$ and $\mu_2 = 1.50$ for $\lambda_2$. Find the distance of the point on the screen closest to the geometric center $O$ (the point on the screen opposite to the perpendicular bisector of the slits) where a bright fringe of $\lambda_1$ exactly coincides with a bright fringe of $\lambda_2$.

A beam of electrons (mass $m$, charge $-e$, initial velocity $v_0 \hat{i}$) is incident on a double-slit apparatus. The slits are separated by a distance $d$ along the y-axis (located at $y = \pm d/2, z=0$). A screen is placed at a distance $D$ from the slits at $x=D$. A uniform longitudinal magnetic field $\vec{B} = B \hat{i}$ is applied in the region between the slits and the screen. Assuming $d \ll D$ and that the transverse velocity of the electrons is small compared to $v_0$, what is the new fringe width $\beta$ of the interference pattern observed on the screen along the y-axis at $z=0$?

Let $\beta_0 = \frac{h D}{m v_0 d}$ be the standard fringe width in the absence of the magnetic field, and $\theta = \frac{e B D}{m v_0}$ be the cyclotron phase angle completed by the electrons during their transit.

A cylindrical vessel is filled with a transparent liquid up to a height $H$. The temperature of the liquid at the bottom of the vessel is maintained at $T_0$, and at the free surface (top) it is maintained at $T_0 + \Delta T$, establishing a steady linear temperature gradient. The volumetric expansion coefficient of the liquid is $\gamma$. The refractive index of the liquid at the bottom is $n_0$. Assume that the refractive index $n$ and density $\rho$ of the liquid are related by the Gladstone-Dale relation, $(n - 1) \propto \rho$. If the vessel is viewed normally from above in air ($n_{\text{air}} = 1$), what is the apparent depth of the bottom of the vessel?

A coherent point light source $S$ emitting light of wavelength $\lambda$ (in $\text{m}$) is attached to the bob of a simple pendulum. The pendulum consists of a small spherical bob of mass $m$ (in $\text{kg}$) and radius $a$ (in $\text{m}$), suspended by a massless string of length $L$ (in $\text{m}$). The source $S$ is positioned at a distance $D'$ (in $\text{m}$) behind a plane containing two narrow slits separated by a distance $d$ (in $\text{m}$). The equilibrium position of the pendulum lies exactly on the central axis of the double-slit system. The pendulum oscillates in a vertical plane parallel to the slit plane, within a medium of viscosity $\eta$ (in $\text{Pa}\cdot\text{s}$). At $t=0$, the pendulum is released from a small initial angular displacement $\theta_0$ (in $\text{rad}$). An interference pattern is observed on a screen placed at a distance $D$ (in $\text{m}$) from the slits. (Assume the initial amplitude of the central maximum is strictly greater than one fringe width). Assuming small angle approximations and underdamped oscillations, what is the time $t$ (in $\text{s}$) after which the amplitude of oscillation of the central maximum on the screen reduces to exactly one fringe width?

A vertical glass capillary tube of inner radius $r = 1.0 \text{ mm}$ is partially immersed in a liquid of refractive index $\mu = 1.5$. The liquid perfectly wets the glass (contact angle $\theta_0 = 0^\circ$). Due to gravity, the meniscus is not perfectly spherical; the exact height of the meniscus edge relative to its apex is measured to be $H = 1.2 \text{ mm}$. A broad, coherent parallel beam of monochromatic light of wavelength $\lambda = 600 \text{ nm}$ (in vacuum) is directed vertically upwards through the liquid column. Consider two specific rays from the incident beam: Ray 1 travels along the central axis and reflects at the meniscus apex. Ray 2 travels at a radial distance infinitesimally close to the glass wall ($x \to r$) and reflects at the extreme edge. Both rays start in phase from a common horizontal reference plane inside the liquid. What is the difference in their accumulated phase, $\Delta \Phi$, evaluated immediately after their respective reflections at the liquid-air interface?

In a Young's Double Slit Experiment, a beam of photons of wavelength $\lambda$ is incident on two slits separated by a distance $d$. The interference pattern is observed on a screen at a distance $D$ ($D \gg d$). To obtain 'which-path' information, a quantum non-destructive detector is placed immediately after the slits to measure the transverse position $y$ of each photon. The measurement has a Gaussian uncertainty profile with standard deviation $\Delta y = a$. According to the Heisenberg uncertainty principle, this measurement imparts a random transverse momentum kick $\Delta p_y$ to the photon, which follows a Gaussian distribution with standard deviation $\sigma_p = \frac{h}{4\pi a}$ (where $h$ is Planck's constant). If the original interference pattern without the detector has a visibility of $1$, what is the visibility $V$ of the fringes on the screen with the detector in place?

(Hint: The convolution of $f(y) = \cos(ky)$ with a Gaussian probability density function $P(y) = \frac{1}{\sqrt{2\pi}\sigma}e^{-y^2/(2\sigma^2)}$ is $\cos(ky)e^{-k^2\sigma^2/2}$)

In a Young's Double Slit Experiment, two identical slits $S_1$ and $S_2$ of width $b$ are separated by a distance $d$ ($d \gg b$). The screen is at a distance $D$ ($D \gg d$). The geometric center of the screen is at $y=0$, with $S_1$ and $S_2$ located at $y = d/2$ and $y = -d/2$ respectively. Two identical transparent rectangular pipes of inner thickness $t$ are placed directly in front of $S_1$ and $S_2$. An ideal gas of density $\rho_0$ flows steadily through both pipes. The velocity profile of the gas across $S_1$ (with local coordinate $y_1 \in [-b/2, b/2]$ measured upwards from the center of $S_1$) is $v_1(y_1) = v_0 \sqrt{1 + \alpha y_1}$. The velocity profile across $S_2$ (with local coordinate $y_2 \in [-b/2, b/2]$ measured upwards from the center of $S_2$) is $v_2(y_2) = v_0 \sqrt{1 - \alpha y_2}$. Assume the flow is incompressible for calculating pressure variations. The refractive index $n$ of the gas depends on its local pressure $P$ as $n = 1 + K P$, where $K$ is a positive constant. Due to the velocity gradients, the diffraction envelopes of both slits are shifted. It is observed that the central maxima of the diffraction envelopes of both $S_1$ and $S_2$ perfectly coincide at the geometric center of the screen ($y=0$). Which of the following expressions gives the correct value of $\alpha$? (Assume small angles, $\sin\theta \approx \tan\theta \approx \theta$).

In a Young's Double Slit Experiment (YDSE), two identical slits emit coherent light of intensity $I_0$ and optical angular frequency $\omega_{opt}$. An electro-optic modulator is placed in front of one of the slits, introducing a time-varying phase shift $\phi(t) = \alpha V(t)$, where $\alpha$ is a positive constant in $\text{rad/V}$ and $V(t)$ is the applied voltage. The voltage is supplied by an ideal $L-C$ circuit (inductor $L$, capacitor $C$). At $t=0$, the capacitor is fully charged to a voltage $V_0$ and the current is zero. Assume the $L-C$ oscillation frequency is much smaller than $\omega_{opt}$. Given the integral identity $\frac{1}{2\pi} \int_0^{2\pi} \cos(x \cos \theta) d\theta = J_0(x)$, where $J_0(x)$ is the zeroth-order Bessel function of the first kind, what is the time-averaged intensity $\langle I \rangle$ at the central point on the screen (where the geometrical path difference is zero) over one complete period of the $L-C$ oscillation?

A linearly polarized plane electromagnetic wave of uniform intensity $I_0$ and wavelength $\lambda$ is incident normally on a perfectly opaque, flat screen lying in the $x$-$y$ plane. The screen has a long, narrow slit of width $a$ parallel to the $y$-axis.

Due to diffraction, the transmitted wave spreads out. In the far-field ($r \gg a$), the time-averaged Poynting vector is directed radially outward from the slit. Its magnitude at a distance $r$ and angle $\theta$ (measured from the $z$-axis in the $x$-$z$ plane) is given by: $|\langle \vec{S} \rangle| = \frac{2 P_0}{\pi r} \cos^2\theta \quad \text{for } -\frac{\pi}{2} \le \theta \le \frac{\pi}{2}$ where $P_0$ is the total transmitted power per unit length of the slit (in $\text{W/m}$).

A perfectly absorbing, rigid semi-cylindrical surface of radius $R$ ($R \gg a$) is placed symmetrically behind the slit (centered on the $y$-axis) to completely intercept the transmitted wave. What is the net radiation force per unit length exerted on this semi-cylindrical surface? (Let $c$ be the speed of light in vacuum)

A point source $S$ and a detector $D$ are placed in a large block of transparent dielectric material of refractive index $n = \sqrt{2}$. They are located at a depth $h$ below the flat horizontal interface between the dielectric and air ($n_{\text{air}} = 1$). The horizontal distance between $S$ and $D$ is $x = 2\sqrt{3}h$. The source emits s-polarized electromagnetic waves (electric field perpendicular to the plane of incidence) of wavelength $\lambda$ in the dielectric. The detector records the interference between the direct wave from $S$ and the wave totally internally reflected from the interface. What is the minimum depth $h$ for which the detector records a local maximum in intensity?

A monochromatic light beam of wavelength $\lambda$ illuminates a single slit of width $a$. The resulting diffraction pattern is observed on a screen placed at a distance $D$ ($D \gg a$). A small rectangular photoelectric detector of width $w$ ($w \ll \frac{\lambda D}{a}$) and length $L$ is placed on the screen parallel to the slit. The work function of the detector material is $\phi$ (where $\phi < \frac{hc}{\lambda}$). When the detector is centered exactly at the central maximum of the diffraction pattern, it records a saturation photoelectric current $i_0$ and a stopping potential $V_0$. When the detector is moved and centered exactly at the first diffraction minimum, it records a saturation photoelectric current $i_1$ and a stopping potential $V_1$. Assuming the quantum efficiency of the detector is constant and independent of intensity, which of the following expressions correctly gives $V_1$ and the ratio $\frac{i_1}{i_0}$?

Two identical uniform metal rods, each of length $L$, cross-sectional area $A$, Young's modulus $Y$, and coefficient of linear thermal expansion $\alpha$, are aligned horizontally. Their outer ends are rigidly clamped to two immovable vertical walls. The inner ends of the rods face each other, leaving a small gap of width $a_0$ ($a_0 \ll L$). This gap acts as a single slit for a diffraction experiment. The inner ends are also connected by an ideal spring of stiffness $k$ (placed outside the path of the light) which has negligible thermal expansion. Initially, the spring is in its relaxed state. A monochromatic light of wavelength $\lambda$ is incident normally on the slit, producing a diffraction pattern on a distant screen. The temperature of the rods is uniformly increased by $\Delta T$. As a result, the angular width of the central diffraction maximum is observed to double. Assuming the angles involved are small, which of the following expressions gives the required temperature increase $\Delta T$?

A monochromatic plane wave of wavelength $\lambda$ (in vacuum) is incident normally on a single slit of width $a$ and thickness $t$. The slit is filled with a transparent medium whose refractive index varies linearly across the slit width as $n(x) = n_0 + \gamma x$, where $x$ is the transverse coordinate from the center of the slit ($-a/2 \le x \le a/2$) and $\gamma$ is a positive constant such that $\gamma t < 1$. Assuming the thin phase screen approximation (ray bending inside the slit is negligible), what is the angular position $\theta$ of the central principal maximum in the Fraunhofer diffraction pattern observed on a distant screen?

An electron (rest mass $m_0$, charge magnitude $e$) is accelerated from rest through a potential difference $V = \frac{4m_0 c^2}{e}$, where $c$ is the speed of light in vacuum. The accelerated electron beam is incident normally on a single slit of width $d$ to produce a Fraunhofer diffraction pattern on a screen at a distance $D$ ($D \gg d$). Let $W_r$ be the true width of the central maximum calculated using relativistic mechanics, and $W_c$ be the width calculated assuming classical (non-relativistic) mechanics. What is the ratio $\frac{W_r}{W_c}$?

Consider a modified single-slit Fraunhofer diffraction experiment. A monochromatic plane wave of wavelength $\lambda$ (wave vector $k = 2\pi/\lambda$) is incident normally on a slit of width $a$ located from $x = -a/2$ to $x = a/2$. Due to a special absorbing filter placed across the slit, the amplitude of the transmitted electric field is not uniform. Instead, it varies spatially as $E(x) = E_0 e^{-\alpha x}$ for $-a/2 \le x \le a/2$, where $x$ is the distance from the center of the slit and $\alpha$ is a positive constant ($\alpha > 0$). Let $I_0$ be the intensity of the diffracted light at the central maximum ($\theta = 0$). Which of the following expressions correctly represents the intensity $I(\theta)$ of the diffracted light at an angle $\theta$ with respect to the central axis?

A telescope with a circular objective lens of diameter $D = 0.122 \text{ m}$ is used to observe two distant point sources emitting light of frequency $f_0 = 6 \times 10^{14} \text{ Hz}$ in a vacuum. The entire setup (sources and telescope) is then completely immersed in a transparent, non-magnetic liquid of dielectric constant $K = 4$. Let $\Delta \theta$ be the new minimum angular separation required to just resolve the two sources according to the Rayleigh criterion, and $f$ be the frequency of the light in the liquid. Which of the following represents the correct values for $\Delta \theta$ and $f$? (Take the speed of light in vacuum as $c = 3 \times 10^8 \text{ m/s}$)

A circular aperture of a telescope is formed by a smart material membrane. Due to a heavily damped boundary condition, the effective diameter of the aperture decreases exponentially with time as $D(t) = D_0 e^{-bt}$, where $D_0$ is the initial diameter in meters ($\text{m}$) at $t=0$ and $b$ is a positive damping constant in inverse seconds ($\text{s}^{-1}$). The telescope is used to resolve two distant point sources emitting light of wavelength $\lambda$ (in $\text{m}$). Assuming the standard Rayleigh criterion applies instantaneously, what is the time-averaged angular resolution limit $\langle \theta \rangle$ of the telescope over the time interval from $t=0$ to $t=1/b$?

Two point objects are placed at a distance $u = 10 \text{ cm}$ in front of a thin convex lens of focal length $f = 20 \text{ cm}$ and a large aperture. An observer's eye is placed at a distance $x = 30 \text{ cm}$ behind the lens to view the virtual images of the objects. The pupil of the eye has a diameter $d = 3.0 \text{ mm}$ and acts as the limiting aperture of the optical system. If the wavelength of light used is $\lambda = 600 \text{ nm}$, what is the minimum separation $y$ between the two objects such that their images are just resolved by the eye? (Assume the paraxial approximation holds for the position of the images).

An unpolarized light ray is launched from air ($n=1$) into a transparent medium with a gradient refractive index $n(z) = \sqrt{3 - \alpha z}$ at $z=0$. The ray is launched at an angle $\theta_0$ with the $z$-axis. At a depth $z=d$, the refractive index is $n(d) = \sqrt{2}$. At this depth, there is a horizontal interface separating the gradient medium from a uniform cladding of refractive index $n_2 = 1$. It is desired that the ray undergoes Total Internal Reflection (TIR) at $z=d$, AND that the reflected ray is completely plane-polarized. Which of the following represents the required launch angle $\theta_0$?

An unpolarized light beam of time-varying intensity $I(t) = I_0 \cos^2(\omega t)$ and wavelength $\lambda$ is incident from vacuum onto a flat glass slab of refractive index $n = \sqrt{3}$. The angle of incidence is exactly the Brewster's angle. The transmitted beam, having a cross-sectional area $A$ normal to its direction of propagation, immediately passes through a polarizing filter. The transmission axis of the filter rotates with time as $\phi(t) = \omega t$, where $\phi=0$ corresponds to the plane of incidence. What is the time-averaged photon flux (number of photons per second) emerging from the polarizing filter? (Assume $h$ is Planck's constant and $c$ is the speed of light)

In an optics experiment, a student determines the refractive index $\mu$ of a transparent medium by measuring the polarizing angle (Brewster's angle) $i_p$ for unpolarized light incident from air. The measured value of the polarizing angle is $i_p = 60^\circ$. If the spectrometer used has a least count of $\Delta i_p = 0.1^\circ$, what is the maximum percentage error in the calculated value of the refractive index?

A plane wavefront of unpolarized light is incident from vacuum ($z < 0$) onto a non-magnetic dielectric medium occupying the region $z \ge 0$. The refractive index of the medium varies with depth $z$ as $n(z) = \sqrt{3 - \alpha z}$ for $0 \le z < 3/\alpha$, where $\alpha = 1 \text{ m}^{-1}$. The incident light is directed such that the reflected light at the interface $z=0$ is completely plane-polarized. A ray of this light enters the medium at the origin $(0,0,0)$. Determine the $x$-coordinate of the ray at the point where its associated wavefront becomes parallel to the $z$-axis.

A wide, uniform, linearly polarized light beam propagates along the $+x$-axis. Its electric field vector is initially aligned with the $+y$-axis. The beam passes through a thin, non-uniform phase plate located at $x=0$. Upon exiting the plate, the rays are deflected in the $xy$-plane such that the ray at a transverse coordinate $y$ makes an angle $\theta(y) = \frac{\pi y}{3 y_0}$ with the $x$-axis. The magnitude of the Poynting vector along the new propagation direction remains a constant $I_0$ (in $\text{W/m}^2$) for all rays.

Immediately after the plate, the light falls on an ideal flat linear polarizer placed in the $yz$-plane. The transmission axis of the polarizer is parallel to the $y$-axis. The polarizer transmits only the component of the electric field parallel to its transmission axis, such that the magnitude of the transmitted Poynting vector is given by Malus's law.

What is the total transmitted power $P$ through a rectangular aperture on the polarizer defined by $-y_0 \le y \le y_0$ and $0 \le z \le h$?

A rigid square conducting loop of side $L$ and total resistance $R$ lies in the $xy$-plane. At $t=0$, it is pulled with a constant velocity $v_0 \hat{i}$ into a region $x > 0$ containing a uniform magnetic field $B_0 \hat{k}$. Mounted on the loop is an optical system consisting of a fixed polarizer and a second polarizer rotating at a constant angular velocity $\omega$. Unpolarized light of intensity $2I_0$ is incident on the system, such that the transmitted intensity is $I(t) = I_0 \cos^2(\omega t)$. A transducer on the loop uses this light to generate an additional uniform magnetic field $B_{int}(t) = \beta I(t) \hat{k}$ strictly confined within the entire area $L^2$ of the loop. Neglecting self-inductance, what is the average electrical power dissipated in the loop as heat during the time it is partially inside the external magnetic field? (Assume the time taken to enter is much larger than $2\pi/\omega$).