Past exam of the physics A course of the University of Cambridge / 2016 / Paper 2 / Section D

Fresnel diffraction is interference during propagation when curvature of the optical path across an aperture matters. It provides a continuous account of fringes near geometric edges and their evolution into the far-field pattern. A screen that is many aperture widths away is not automatically in the Fraunhofer diffraction limit; wavelength also enters the comparison.

For plane-wave illumination of a transmission $h(\xi ,\eta )$ at $z=0$, the Huygens-Fresnel principle in the scalar paraxial approximation givesIt follows by expanding the propagation distance as $z+[(X-\xi {)}^2+(Y-\eta {)}^2]/(2z)$ while treating the amplitude prefactor as slowly varying. Small propagation angles justify dropping higher path terms. Finite coherent illumination, vector edge fields and very small apertures can require a more complete theory. The Fresnel number $N_F=a^2/(\lambda z)$, with a clearly declared aperture scale $a$, measures whether the aperture's quadratic phase is significant. When it is negligible, the remaining linear phase produces the aperture Fourier transform.^[ref]

Define the optical Fresnel integralsTheir limiting values are $C(\infty )=S(\infty )=1/2$, with odd continuation to negative arguments. The Cornu spiral plots $(C(u),S(u))$: the field contributed by a finite interval is proportional to the chord between its endpoints. Intensities require squared chord lengths; summing zone intensities would lose the cancellation mechanism.

For an opaque straight edge transmitting $\xi >0$ and all $\eta$, let $v=X\sqrt{2/(\lambda z)}$, so $X>0$ is the geometrically illuminated side. Integrating in the unrestricted transverse direction and using the half-line in the other giveshenceFar into the shadow, $I\to 0$; far into the illuminated region, $I\to I_{\mathrm{in}}$. At the geometric shadow boundary, $I(0)=I_{\mathrm{in}}/4$, not half the incident intensity: the field is halved there. On the illuminated side, alternating constructive and destructive contributions give bright and dark fringes whose contrast declines away from the edge. Their positions scale with $\sqrt{\lambda z}$ and are not equally spaced. A finite detector averages fine fringes and tends to recover the geometric-optics step. In a controlled experiment, translate a straight edge through a broad collimated beam, scan a small photodetector across the screen and normalize against an unobstructed reference; vary $z$ to test the square-root scaling. Source size, spectral bandwidth, edge roughness and detector aperture wash out contrast and must be distinguished from a failure of the propagation law.

For a rectangular aperture of full widths $a,b$, defineThe two transverse integrals separate:andA long slit leaves only one varying Fresnel factor. For a square centred on axis, $a=b$ and $w=(a/2)\sqrt{2/(\lambda z)}$ giveOpposite edge contributions overlap; changing aperture size or propagation distance changes their relative phase. Consequently a geometrically illuminated centre can brighten or darken, rather than remaining a fixed top-hat plateau. The original rectangular-aperture map shows fringes parallel to both edges and their crossing structure. Outside the geometric rectangle the wave also extends into the shadow.

At large Fresnel number an interior region approaches geometric illumination while edge fringes remain on a scale $\sqrt{\lambda z}$. At intermediate values the opposite-edge fringes interact across the whole aperture image. In the far field, the formula tends to the product of sinc-squared factors,The far-field lobe scale $\lambda z/a$ grows linearly with $z$, unlike the local near-edge scale. The transition is controlled by the neglected phase, not an abrupt physical boundary.

To test the main effect, record normalized profiles at several distances, calculate the Fresnel integral with the measured aperture dimensions and compare fringe positions before fitting a free intensity scale. Use the same transverse coordinate origin and check that the incident beam is sufficiently uniform. Babinet's principle relates complementary aperture and obstruction fields through $U_a+U_o=U_{\mathrm{free}}$; their near-field intensities are not generally equal, because the unobstructed field remains present and interferes with the diffracted field. This is why a far-field obstruction argument cannot simply be carried over as equality of Fresnel intensities.

The free electron model replaces the detailed periodic potential by a constant background, treating conduction electrons as mobile particles with approximately parabolic dispersion $E={\hslash }^2{k}^2/(2m)$. It is useful to distinguish the classical Drude model from the quantum free electron gas. They share a simple transport picture, but differ fundamentally in occupation statistics.

With a momentum-relaxation time $\tau$, both lead in a simple isotropic one-band approximation toThis explains approximately Ohmic longitudinal conduction and a carrier-density scale from the Hall effect. The relaxation time is phenomenological: introducing it does not itself predict scattering from defects or phonons. A four-probe resistance measurement separates the sample's longitudinal response from current-contact voltage drops; field and current reversals isolate the transverse signal.

The classical model incorrectly assigns roughly $3N{k}_B/2$ of electronic heat capacity, far too large in a degenerate metal. Fermi-Dirac statistics fixes this: most occupied states cannot change occupation after a small thermal perturbation because nearby low-energy states are already filled. Only a fraction of order $k_{B}T/E_F$ near the Fermi energy participates. For $T\ll T_F$,The quantum model also gives a finite Fermi velocity even at zero temperature and Pauli paramagnetism. Under predominantly elastic scattering, its electronic heat and charge conductivities obey the Wiedemann-Franz law,This is a quantum success; classical equipartition does not give the correct small heat capacity and Lorenz number together.

Its limitations are equally instructive. Without the periodic potential it has no band gaps, so it cannot distinguish a filled-band insulator from a metal or explain semiconductor doping. The simplest single-electron Hall formula cannot explain positive Hall coefficients; band curvature and electron holes, or a multiband transport treatment, are needed. It neglects much Fermi-surface anisotropy and detailed optical interband absorption. Noninteracting electrons alone do not explain ferromagnetism or superconductivity, and lattice thermal behavior needs phonons. The nearly-free electron model, which allows reciprocal-lattice Fourier components of the potential to mix plane waves, provides a controlled first correction near Bragg planes.

Its main success is a simple quantitative baseline for degenerate conduction electrons; its main failure is treating the crystal and interactions as too featureless. A fit of conductivity alone is not a test of the whole model: compare heat capacity, Hall sign, optical response and temperature dependence using independently constrained parameters.

In an electrical insulator, lattice vibrations usually carry most heat. A small temperature gradient produces an imbalance between phonons moving in opposite directions. In an isotropic kinetic approximation,where $c_V$ is heat capacity per unit volume, $v_s$ an appropriate acoustic speed and ${\ell }_{\mathrm{tr}}$ the transport mean free path. The factor one third is the angular average of the squared direction cosine. With several branches and frequency-dependent scattering, replace this estimate by the corresponding weighted mode integral. A short lifetime matters for conductivity only insofar as it relaxes heat current.

For a clean three-dimensional crystal at $T\ll {\Theta }_D$, the Debye heat capacity varies as $T^3$. If diffuse boundary scattering fixes ${\ell }_{\mathrm{tr}}$ near a specimen dimension, then ${\kappa }_{\mathrm{ph}}\propto T^3$. The size dependence is a test of that boundary-limited regime. Specular surfaces, defects and small specimens can change the simple law; at ballistic scales a local conductivity description itself needs care.

As temperature rises, more phonons are excited, but phonon-phonon scattering shortens their transport path. Normal processes conserve total crystal momentum, whereas an Umklapp process satisfies ${\mathbf{q}}_1+{\mathbf{q}}_2-{\mathbf{q}}_3=\mathbf{G}\ne 0$ with reciprocal vector $\mathbf{G}$ and can relax directed heat flow. Low-temperature Umklapp scattering is strongly suppressed; its increasing importance often produces a conductivity maximum followed by a decline. Isotope and impurity scattering lower or broaden that maximum.

At sufficiently high temperature, $c_V$ approaches its classical constant while a typical anharmonic Umklapp rate grows approximately as $T$. Under that further approximation,Thus a clean crystalline insulator commonly shows a low-temperature rise, a peak, then a high-temperature fall. This is not a universal curve for every insulator: glasses, strong disorder, multiple scattering mechanisms, radiative heat transfer and structural transitions can give different behavior.

A steady-state bar experiment supplies a concrete test. Inject known heating power, measure the small temperature difference between two separated sensors and infer $\kappa =PL/(A\Delta T)$ after correcting for heat leaks. Keep the gradient small enough that a single representative temperature is meaningful, calibrate sensor contact effects and subtract radiative/background losses. Repeat with different widths or isotope/defect content to distinguish boundary limitation from intrinsic scattering. A power-dependent result may reflect a changing temperature profile rather than a new microscopic law.

A p-n junction brings p-type and n-type regions into contact. Majority-carrier diffusion initially leaves exposed ionized acceptors and donors, producing a depletion region and a built-in potential. At thermal equilibrium the electrochemical potential is constant: drift and diffusion currents cancel. The internal barrier is not a freely extractable battery voltage.

For a nondegenerate abrupt homojunction,Here $V>0$ denotes forward bias and the depletion approximation is assumed. forward bias lowers the barrier and injects minority carriers; reverse bias widens the depleted region and suppresses ordinary majority injection. In the ideal diffusion regime,Recombination can change the ideality factor; series resistance, high-level injection, leakage and breakdown limit this simple Shockley diode equation. The original junction theory explicitly treats both electron and hole currents, which is essential to carrier injection and transistor action.^[ref]

As a rectifying diode, the junction conducts strongly in one bias direction and weakly in the other before breakdown. This permits power rectification, signal detection and electronic switching. Stored minority charge affects turn-off time, so a static exponential current-voltage relation alone does not specify switching speed. A junction capacitance $C_j\simeq {ϵ}_{s}A/W$ varies with reverse bias and can be used in a voltage-controlled capacitor; heavily doped or high-field junctions can instead exploit controlled tunnelling or avalanche breakdown.

In a photodiode, absorbed photons generate electron-hole pairs and the depleted-region field assists collection. reverse bias widens the collecting region and can improve speed, but adds dark-current noise and breakdown constraints. In a solar cell, the same carrier generation supplies current against an external voltage. An ideal illuminated characteristic is $I=I_s(e^{qV/(k_{B}T)}-1)-I_{\mathrm{ph}}$: at open circuit the forward diode current balances photocurrent, while useful power is taken at an intermediate operating point. Neither short circuit nor open circuit delivers power to a load.

A forward-biased light-emitting diode injects electrons and holes that can recombine radiatively. Efficient emission favors suitable direct-gap materials and carrier confinement; the characteristic photon energy is near the semiconductor band gap. nonradiative recombination, escape losses, series resistance and heating limit efficiency. Two coupled junctions also form a bipolar junction transistor: an emitter injects minority carriers through a thin base, and a reverse-biased collector collects them. Useful current gain depends on efficient injection and limited base recombination, not simply on joining any three arbitrary regions.

These functions are experimentally distinguished by measuring dark and illuminated current-voltage curves, bias-dependent capacitance, transient switching or optical emission, with controlled temperature and current limits. The common mechanism is bias-controlled carrier injection or collection at a depletion barrier; the device geometry and operating mode determine which function is useful.