Past exam of the laser science and quantum information processing course of the University of Oxford / 2015 / 4

For bulk electric-dipole second-harmonic generation, the crystal must lack inversion symmetry and have a nonzero relevant component of its second-order nonlinear optical susceptibility. It should transmit both pump and harmonic, tolerate the required intensity, and allow phase matching by suitable birefringence or another phase-compensation mechanism. A large nonlinear coefficient, low optical loss and good optical quality are desirable. Noncentrosymmetry permits a response but does not guarantee one in every chosen geometry.

In the undepleted plane wave limit, contributions from successive layers give a harmonic amplitude proportional to ${\int }_0^L{e}^{i\Delta kz}dz$, where $\Delta k=k_{2\omega }-2k_{\omega }$. HenceTune orientation, temperature and pump polarization to maximize the nonlinear coupling and cancel phase mismatch. For continuous operation, an impedance-matched enhancement optical resonator, or intracavity doubling, raises the circulating intensity. Focus the Gaussian beam to balance intensity against diffraction and spatial walk-off; arbitrarily tight focusing shortens the useful overlap. Antireflection surfaces, stable resonance locking and efficient heat removal reduce avoidable loss and drift.

For a fundamental electric field in the $xy$ plane, $E_z=0$. The specified tensor then givesWriting $E_x=E_0\cos \varphi$ and $E_y=E_0\sin \varphi$ yieldsThe source electric polarization is therefore along $z$ and is largest along the in-plane diagonals. A radiated wave also needs a transverse component of this source: propagation along $z$ would not generate a propagating $z$-polarized output. Propagation in the $xy$ plane provides the simple strictly transverse example. For a tilted extraordinary wave, the electric displacement, rather than generally the entire electric field, is perpendicular to the wave vector; the tensor source must be projected onto that wave's allowed polarization. This distinction matters when using the type-I geometry below.

Type-I phase matching requires $n_{e,2\omega }(\theta )=n_{o,\omega }$, because both pump photons are ordinary and the harmonic is extraordinary. Apply the direction-dependent index equation to the harmonic, not to the pump. Solving givesFor the ruby-laser data this is $0.626210$, whereas the shorter-wavelength data give $0.958873$. ThereforeThe second calculation uses the supplied $257\mathrm{nm}$ indices as rounded harmonic data: the exact half-wavelength of $515\mathrm{nm}$ is $257.5\mathrm{nm}$. The index dispersion between those two ultraviolet values was not supplied.

For an extraordinary wave, its Poynting vector makes an angle ${\theta }_S$ with the optic axis satisfying $\tan {\theta }_S=(n_o^2/n_e^2)\tan \theta$. Thus the spatial walk-off angle relative to the wave vector isBoth angles refer to the harmonic. The ordinary pump does not have this angular walk-off. For beam radius $w_b$, the approximate overlap length is $w_b/\tan \rho$, so the shorter-wavelength geometry has about $2.19$ times the walk-off-limited interaction length.^[ref]

To compare coupling, take propagation azimuth $\phi$ and ordinary unit polarization $(-\sin \phi ,\cos \phi ,0)$. Its quadratic source is $P_z=-{ϵ}_0{d}_{36}{E}_0^2\sin 2\phi$. Projection onto an extraordinary displacement direction with $z$ component proportional to $\sin \theta$ gives $|d_{\mathrm{eff}}|\propto |d_{36}\sin \theta \sin 2\phi |$, up to the small anisotropic normalization correction. The optimal azimuth is $45^{\circ }$. At equal beam area, interaction length, input power and approximately equal nonlinear coefficients, the shorter pump has both stronger angular coupling and the favorable frequency-squared factor:Here $\eta =P_{2\omega }/P_{\omega }$ in the small-conversion, equal-area model, with harmonic index phase-matched to the pump. Its smaller spatial walk-off further favors the $515\mathrm{nm}$ pump in a walk-off-limited geometry. Equal input power alone does not determine an absolute efficiency ranking: ultraviolet absorption, wavelength-dependent $d_{36}$, beam size, crystal length and coatings are unspecified. For example, a sufficiently strongly absorbing $257.5\mathrm{nm}$ output can reverse the ideal ranking. The stated preference is conditional on comparable transparent crystals and comparable focusing, not a prediction independent of those parameters.