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For a vertical interband transition, the crystal momentum is effectively unchanged because the photon momentum is small. A stimulated downward transition needs an occupied upper state and an empty lower state, with factor $f_c(E_2)[1-f_v(E_1)]$; upward photon absorption has factor $f_v(E_1)[1-f_c(E_2)]$. The two matrix elements agree by microscopic reversibility. Their difference is thereforeA common slowly varying transition-strength factor is implicit. Broadening replaces the sharp transition by a convolution; multiple valence bands contribute separate terms.

For one isotropic pair of parabolic energy bands, writeThus $E_2-E_1=E_g+{\hslash }^2{k}^2/(2m_r)$. Including the usual twofold spin degeneracy, the number of paired states per volume inside a sphere of radius $k$ is $k^3/(3{\pi }^2)$. Differentiating with respect to transition energy $E$ gives the three-dimensional parabolic interband joint density of statesFor density per angular frequency, $g_j^{(\omega )}(\omega )=\hslash g_j^{(E)}(\hslash \omega )$. Stating the density convention avoids a missing factor of $\hslash$; other degeneracies multiply the formula.

At a common carrier temperature, the Fermi-Dirac distribution is a strictly decreasing function of $E-F$. The direct-gap semiconductor gain condition follows:The lower inequality denotes a transition above the ideal band edge, where the displayed joint density of states is nonzero. Thus quasi-Fermi-level splitting must exceed the transition photon energy. Equality is transparency, rather than net amplification; at zero temperature an allowed transition must also lie within the occupied-electron and empty-valence windows.

Put $b=\mathcal{E}{e}^{i(kr-\omega t)}$, allowing $\mathcal{E}$ to be complex. The Hermitian electric field is $\hat{E}=b\hat{a}+b^{\ast }{\hat{a}}^{†}$. The annihilation operator and creation operator change photon number, so their diagonal matrix elements in a Fock state vanish. Since $\hat{a}{\hat{a}}^{†}=\hat{n}+1$ and ${\hat{a}}^{†}\hat{a}=\hat{n}$,The terms containing ${\hat{a}}^2$ or ${\hat{a}}^{†2}$ connect orthogonal number states and contribute zero. Even as $n$ grows, there is no definite oscillating mean electric field; its quadrature fluctuations grow instead. A Fock state can represent fixed field energy, but not a classical sinusoidal wave of prescribed wave phase. This is distinct from representing a classical field with a randomly averaged phase.

The incident intensity radius is $w_L=2.5\mathrm{mm}$, half the quoted diameter. For an ideal collimated Gaussian beam focused by a thin lens, the diffraction-limited waist isThe incident Rayleigh range $\pi w_L^2/{\lambda }_0\simeq 24.5\mathrm{m}$ is much larger than $f$, justifying this approximation. The pulse energy is $U=P_{\mathrm{av}}/R=0.25\mu \mathrm{J}$. Taking its duration as an effective energy-to-peak-power width gives $P_{\mathrm{pk}}\simeq U/\tau =7.14\times 10^5\mathrm{W}$. Since $I(r)=I_0{e}^{-2r^2/w_0^2}$ has integrated power $I_0\pi w_0^2/2$,If $350\mathrm{fs}$ denotes the temporal intensity full width at half maximum of a Gaussian pulse, the time integral instead gives $P_{\mathrm{pk}}=2\sqrt{\ln 2}U/(\sqrt{\pi }\tau )$, and $I_{\mathrm{pk}}=1.03\times 10^{13}\mathrm{W}\mathrm{c}{\mathrm{m}}^{-2}$. The pulse shape is unspecified, so the estimate's order-one convention is explicit. Aberration, clipping and propagation through glass were excluded from this calculation as stipulated.

A laser relaxation oscillation is the damped exchange between stored upper-level excitation and intracavity photon number after a small perturbation of a lasing steady state. Extra photons increase stimulated depletion of the upper level; the resulting smaller gain then reduces the photon population. Pumping subsequently replenishes the inversion, so the two populations overshoot their equilibrium values before damping settles them.

Laser spiking is the corresponding large-signal sequence of intense output pulses, often during switch-on. Excitation first accumulates while the optical field is small. Once gain exceeds loss, rapid stimulated emission releases this stored energy in a spike, depleting the inversion below threshold. Recovery can produce further spikes. This is distinct from an externally imposed Q-switching cycle, although both draw on stored excitation.

Many solid-state gain media have upper-state lifetimes far longer than the cavity photon lifetime. The optical polarization relaxes rapidly, leaving the two slow rate-equation variables of a class-B laser. A long-lived population reservoir and a much faster photon response make underdamped recovery and pronounced startup spiking common. The criterion and frequency follow from the linearization in part c; they are not universal consequences of the word “solid-state”.

The upper-state density is clamped near threshold once the laser operates, so excess pumping mainly increases photon density and output power. For approximately pump-independent absorption efficiency, $r=P_p/P_{p,\mathrm{th}}$, and both $P_{\mathrm{out}}$ and $f_{\mathrm{ro}}^2$ should rise approximately linearly with $P_p-P_{p,\mathrm{th}}$.

The left ordinate is the square of the frequency, in ${\mathrm{kHz}}^2$, rather than frequency in kHz. Reading the original triangles gives a threshold near $12$–$13\mathrm{mW}$ and a slope about $1.0{\mathrm{kHz}}^2/\mathrm{mW}$. A representative straight-line fit to visually read marker centers gives $P_{p,\mathrm{th}}\simeq 12.2\mathrm{mW}$ and $A=d{f}_{\mathrm{ro}}^2/d{P}_p\simeq 1.02\times 10^6{\mathrm{Hz}}^2/\mathrm{mW}$. ThereforeThe displayed precision is limited by graph reading and scatter; a lifetime around $0.15\mu \mathrm{s}$ is the meaningful result. The output squares have a compatible threshold and an approximately constant slope above it.

The round-trip time isIf mirror transmission is the only appreciable passive loss, energy decays by $R_1{R}_2=e^{-t_{\mathrm{rt}}/{\tau }_c}$ per round trip. For $R_1=1$,a power reflectance of roughly 65%. The exact exponential is preferable to replacing $1-R_2$ by $t_{\mathrm{rt}}/{\tau }_c$, since the inferred round-trip loss is not tiny. If distributed fibre loss is appreciable, the lifetime instead fixes the total round-trip retention; without a separate loss measurement, the bare mirror reflectance is not uniquely determined and is at least this retention factor.

The total excitation operator $\mathcal{N}=a^{†}a+({\sigma }_z+1)/2$ commutes with the Hamiltonian. Each sector with $\mathcal{N}=n+1\ge 1$ is therefore spanned by $|e,n⟩,|g,n+1⟩$, and no term connects it to a sector of different excitation number.

In that ordered basis, with the source's Rabi detuning $\Delta =\omega -{\omega }_0$,The first term is the common sector energy; the remaining diagonal entries record the bare Rabi detuning and the off-diagonal terms exchange the excitation.

The isolated ground sector of the Jaynes-Cummings model is an important exception to a statement that the entire Hilbert space consists of such pairs: $|g,0⟩$ has no partner and is annihilated by both interaction terms. It is a one-dimensional invariant sector with energy $-\hslash {\omega }_0/2$. The full decomposition is this one-dimensional sector plus all the two-dimensional sectors above.

Subtract the common sector energy and call the remaining eigenvalue $\hslash \lambda$. Its characteristic equation isThe dressed-state energies are consequentlyOn resonance their splitting is $2\hslash |g|\sqrt{n+1}$; far off resonance the eigenstates approach the uncoupled quantum states with small dispersive energy shifts. The eigenvalue splitting is an angular-frequency difference ${\Omega }_n$, so an ordinary frequency would be ${\Omega }_n/(2\pi )$.

Write the optical system's spectral response as $A(\omega )e^{i\varphi (\omega )}$. Expanding its spectral phase around the pulse's central angular frequency givesThe group delay is ${\varphi }^{(1)}=d\varphi /d\omega$, with units of time. The linear spectral phase term delays the pulse envelope by this amount. The group delay dispersion is ${\varphi }^{(2)}=d^2\varphi /d{\omega }^2$, with units of time squared. It makes the group delay frequency dependent: $t_g(\omega )\simeq {\varphi }^{(1)}+{\varphi }^{(2)}(\omega -{\omega }_0)$. Positive group delay dispersion delays higher frequencies more and gives positive chirp. A nonzero quadratic spectral phase broadens an initially transform-limited pulse; applied to an already chirped pulse, a suitable opposite sign can compress it.

In chirped pulse amplification, an ultrashort seed pulse first passes through a dispersive stretcher, often a grating-based system. This correlates frequency with arrival time and increases pulse duration while approximately preserving pulse energy, thereby lowering peak intensity. An optical amplifier can then increase pulse energy without exposing the gain medium to the original short pulse's enormous peak intensity, reducing optical damage and unwanted nonlinear phase accumulation. A final dispersive compressor, typically using gratings, cancels the accumulated spectral phase to recover a short, high-energy pulse. The essential components are therefore a short-pulse source, stretcher, amplifier and matched compressor; spectral gain bandwidth and higher-order dispersion limit the achievable compression.

A time delay $d$ multiplies the spectral field by $e^{i\omega d}$. The bright-port transfer function is thereforeThe spectral transmission maxima have angular frequency spacing $\Delta \omega =2\pi /|d|$. For a strongly chirped pulse the local frequency-time mapping is $\omega (t)={\omega }_0+2bt$. Thus the filter's adjacent transmitted frequency bands emerge at times separated byas obtained directly from the temporal fields. The spectral transfer function itself is exact; mapping its features directly onto time requires a sufficiently strong chirp and overlapping envelopes.

As $|d|$ approaches the pulse duration, the two envelopes overlap only partly and the small-delay formula fails. For $|d|\gg \tau$, the selected port contains two separate replicas of field amplitude $E_0/2$, each with one quarter of the input pulse energy. Their combined energy is half the input; the other output port carries the remainder. There is no temporal overlap or corresponding time-resolved interference. A sufficiently resolving spectrometer can still reveal fine spectral fringes, whose spacing decreases as $1/|d|$.

Take ${\sigma }_z|e⟩=|e⟩$ and ${\sigma }_z|g⟩=-|g⟩$. In the resonant Jaynes-Cummings model, the interaction connects only $|e,n⟩$ and $|g,n+1⟩$. Both have free energy $\hslash \omega (n+1/2)$, and their interaction matrix is $\hslash g\sqrt{n+1}{\sigma }_x$. ThereforeExponentiating this two-state block and applying it to $|e,n⟩$ yieldsThe populations undergo Rabi oscillations at angular frequency $2g\sqrt{n+1}$. The common phase in this fixed-$n$ block does not affect its probabilities, but it must be retained when blocks with different photon numbers are superposed.

One implementation uses type-II phase matching for spontaneous parametric down-conversion in a nonlinear birefringent crystal pumped by a coherent laser. A pump photon produces two lower-frequency photons of orthogonal polarizations, subject to energy and momentum phase matching. Collect two spatial modes in which the alternatives “horizontal photon in Alice's mode and vertical photon in Bob's mode” and the reversed assignment are indistinguishable. Compensation crystals, filters and matched optical paths erase polarization-dependent timing and spectral information. The amplitudes then add coherently to give a state proportional to $|HV⟩+e^{i\phi }|VH⟩$; a birefringent phase element tunes $\phi$.

The essential requirement is coherent indistinguishability of the two assignments. Producing two orthogonally polarized photons with information revealing which assignment occurred would instead give a mixture without the required polarization entanglement.

Q-switching stores excitation in a pumped laser gain medium while preventing oscillation, then rapidly changes the optical cavity from high loss, or low quality factor, to low loss, or high quality factor. The initially suppressed stimulated field can then grow rapidly from a seed and deplete a large population inversion in a short pulse. The resulting peak power is much higher than for continuous operation at the same average pumping rate. This is energy storage and release, rather than the phase locking of many longitudinal modes used in mode locking.

An active implementation uses a Pockels cell and a polarizer. An applied voltage makes the cell change the intracavity polarization, causing the polarizer to introduce sufficient loss to prevent lasing while pumping builds the inversion. Rapidly changing the voltage removes that loss and opens the high-Q state. The switching time should be short relative to the pulse build-up and depletion times. A saturable absorber provides an alternative passive Q switch: its absorption decreases as the intracavity intensity rises.

The Q-switched energy extraction efficiency in the ideal loss convention of ii isThe first factor reflects the definition of inversion: one stimulated transition reduces the upper population and increases the lower population, changing $N^{\ast }=N_2-(g_2/g_1)N_1$ by $1+g_2/g_1=\beta$. Thus one emitted photon consumes $\beta$ units of inversion. The second factor accounts for the inversion left after the pulse; finite pumping above threshold does not generally extract all the initially stored inversion. This $\eta$ is not the wall-plug efficiency and does not include the energy cost of pumping or the factor $\zeta$ for imperfect output extraction.

Write $N=N^{\ast }$, $N_i=N_i^{\ast }$ and $N_f=N_f^{\ast }$ during this derivation. Eliminating the stimulated term between the two laser rate equations gives the photon balanceIntegrate over the pulse, taking negligible initial and final stored photon densities:The photons lost from the cavity carry total energy $\hslash {\omega }_L{V}_c\int n/{\tau }_{c}dt$. Since $f_c{V}_c=V_g$,This identifies the supplied energy formula under the ideal extraction convention that all photon loss is useful output. If only a fraction $\zeta ={\kappa }_{\mathrm{out}}/{\kappa }_{\mathrm{total}}$ of the loss exits the output coupler, the measured output energy is $\zeta E$. The rate equations by themselves do not distinguish useful output from parasitic loss.

Extracting 95 percent of the maximum available inversion energy means $\eta =0.95/\beta$, or $x=\beta \eta =0.95$. Substitution in ii givesThe degeneracy factor cancels because the target is a fraction of the maximum energy $N_i^{\ast }{V}_g\hslash {\omega }_L/\beta$, rather than a fraction of $N_i^{\ast }{V}_g\hslash {\omega }_L$ without that factor.

During the pulse, the photon density reaches its maximum when $\dot{n}=0$ with $n>0$, so $N^{\ast }=N_{\mathrm{th}}^{\ast }$. Evaluating i at this inversion givesThe ideal instantaneous output power is $P(t)=V_c\hslash {\omega }_{L}n(t)/{\tau }_c$. ThereforeFor $r\gg 1$, almost all the extractable energy is stored briefly as intracavity photons, and $P_{\max }\simeq E_{\max }/{\tau }_c$. The leading rise while the inversion remains near its initial value is $n\propto e^{(r-1)t/{\tau }_c}$, with short growth time ${\tau }_c/(r-1)$. Depletion then takes the inversion below threshold. At late times $N^{\ast }\simeq N_f^{\ast }$ andwhich approaches a tail with decay time ${\tau }_c$ at large $r$. The pulse is consequently asymmetric, with a rapid rise and a slower approximately exponential trailing edge. The negligible-seed build-up delay depends on the seed strength. The pumping and spontaneous population decay have been neglected over the short pulse; parasitic cavity loss multiplies useful power by the output fraction described in ii.

Let $|g⟩,|e⟩$ be the atomic levels, ${\sigma }_{+}=|e⟩⟨g|$, and $a$ the annihilation operator of the single field mode. Choose the joint ground energy to be zero. The resonant Jaynes-Cummings model isThe first term counts field photons, the second the atomic excitation, and the interaction exchanges one photon for one atomic excitation. A phase redefinition makes $g$ real and positive. Resonance means $\omega ={\omega }_{eg}$; the total excitation $\mathcal{N}=a^{†}a+|e⟩⟨e|$ is conserved. Adding an overall constant gives the equivalent convention using ${\sigma }_z/2$ and a zero-point field term.

The model follows from the electric-dipole interaction and the rotating-wave approximation, retaining the energy-conserving terms while dropping $a{\sigma }_{-}$ and $a^{†}{\sigma }_{+}$. We assume a single quantized monochromatic mode and negligible cavity leakage and atomic relaxation over the time considered. With this convention, the resonant population Rabi angular frequency in the one-excitation subspace is $2g$.

Discarding the photon-number result gives a classical mixture of the two outcome branches. The atomic state, in the basis $(|e⟩,|g⟩)$, isIt is also the reduced state before measuring the field; a nonselective operation on the field does not change the atomic reduced density matrix. The atom is not in a coherent superposition with these populations.

For an atomic energy gap $\hslash \omega$, the thermal population ratio is $p_e/p_g=e^{-\hslash \omega /(k_{B}T)}$. Matching it givesA finite positive temperature exists when $0<p_e<p_g$, with $T\to 0^{+}$ at $p_e=0$ and $T\to +\infty$ as $p_e\to 1/2$ from below. Equal populations correspond to zero inverse temperature. If $p_e>1/2$, the expression is a negative effective temperature for a bounded two-level system, not an equilibrium state at any ordinary positive temperature. For a concrete counterexample to an unrestricted positive-temperature interpretation, at $t=0$ the atom is certainly excited, whereas every positive-temperature Gibbs distribution has $p_e\le 1/2$. The fully excited endpoint is the formal limit $T\to 0^{-}$. Matching one density matrix does not establish genuine thermalization with a bath.

In a direct band gap semiconductor, the small photon momentum makes optical transitions approximately vertical in wave vector. Their energy isThe joint density of states per unit volume per transition energy, including the supplied spin counting, isFor an upper electron state of occupation $f_c$ and a lower electron state of occupation $f_v$, stimulated emission has occupation factor $f_c(1-f_v)$; absorption has factor $f_v(1-f_c)$. Their difference is exactly $f_c-f_v$. Multiplication by the joint density of states therefore givesThe proportionality suppresses the optical transition matrix element and other smoothly varying optical prefactors. It assumes parabolic isotropic energy bands, an allowed direct transition, and negligible linewidth and many-body corrections. Here $f_v$ counts electrons, not electron holes.

Assume equal excess electron and electron hole number densities $N_t$, with no additional doping imbalance. At zero temperature, both occupied carrier spheres have the same Fermi wavevector $k_F$ because their spin counting is the same:For $k<k_F$, the conduction band is occupied and the valence band has electron holes, so $f_c-f_v=1$. Beyond $k_F$ the occupations reverse and the medium absorbs. Thus the upper edge of the positive-optical gain interval is the supremumEquivalently $E_{\mathrm{Fc}}=E_g+{\hslash }^2{k}_F^2/(2m_c^{\ast })$ and $E_{\mathrm{Fv}}=-{\hslash }^2{k}_F^2/(2m_v^{\ast })$. With the supplied effective masses,Using the vacuum wavelength $\lambda =hc/(\hslash \omega )$ reverses the endpoints:These boundaries are idealized and open: the lower-energy edge has zero joint density of states, and the upper edge is the zero-temperature occupation discontinuity, not a finite-width gain peak. The calculation gives positive material optical gain, not the stronger requirement of exceeding the losses at laser threshold.