Notation, Models, and Checks

Source conventions

The core is region 2, with index \(n_2\) and full thickness \(t\) or \(t_g\). Region 1 is the cover and region 3 the substrate. The guide runs along \(z\). We write \(k_0=2\pi/\lambda_0\), \(N=n_{\mathrm{eff}}=\beta/k_0\), and use the book’s time dependence \(e^{i\omega t-i\beta z}\). A guided mode satisfies \(\max(n_1,n_3)<N<n_2\).

The bounce angle \(\theta\) is measured from the guide plane, so \(\beta=k_0n_2\cos\theta\). The incidence angle from the normal is \(\phi=90^\circ-\theta\). These angles must not be interchanged. \(\Delta n=n_2-n_3\) is an index difference; it is not the relative index difference sometimes also denoted \(\Delta\).

Exact slab cutoffs

Take \(n_3\ge n_1\); interchange the cover and substrate labels if necessary. The transverse and evanescent constants are

\[h=k_0\sqrt{n_2^2-N^2},\quad p=k_0\sqrt{N^2-n_3^2},\quad q=k_0\sqrt{N^2-n_1^2}.\]

For TE modes the branch-safe eigenvalue equation is

(1)\[ht=m\pi+\tan^{-1}(p/h)+\tan^{-1}(q/h),\qquad m=0,1,\ldots.\]

At cutoff \(N\to n_3\), giving

(2)\[t_{c,m}^{\rm TE}= \frac{\lambda_0}{2\pi\sqrt{n_2^2-n_3^2}} \left[m\pi+ \tan^{-1}\sqrt{\frac{n_3^2-n_1^2}{n_2^2-n_3^2}}\right].\]

For TM replace \(p\) by \((n_2^2/n_3^2)p\) and \(q\) by \((n_2^2/n_1^2)q\) in the phase equation. At cutoff only the cover phase remains, so multiply the square root in the arctangent by \(n_2^2/n_1^2\).

Count only orders satisfying \(t>t_{c,m}\). An order exactly at cutoff has an infinite evanescent extent and is not a normalizable bound mode. Counts here are per polarization; a TE and a TM fundamental are two vector modes.

For a strongly asymmetric, weak-index-step guide, \(q/h\gg1\) and \(n_2^2-n_3^2\simeq2n_2\Delta n\). Consequently

(3)\[\Delta n_{c,m}\simeq \frac{(2m+1)^2\lambda_0^2}{32n_2t^2}.\]

This approximation is appropriate to Problems 2.1–2.3. It should not replace the exact cutoff when the question supplies a general asymmetric structure.

Power, gain, and semiconductor conventions

The attenuation coefficient \(\alpha\) acts on power: \(P(z)=P(0)e^{-\alpha z}\). The corresponding field amplitude decays as \(e^{-\alpha z/2}\). Thus loss is \((10/\ln10)\alpha=4.3429448\alpha\) dB per matching length unit. Natural logarithms appear in propagation and mirror losses; base-10 logarithms convert power ratios to decibels.

For a laser cavity the modal gain is \(\Gamma g\), with \(\Gamma\simeq d/D\) in the book’s simple confinement model. The intensity reflectivities are \(R_1,R_2\), and

(4)\[\alpha_m=\frac{1}{2L}\ln\frac{1}{R_1R_2},\qquad \Gamma g_{\rm th}=\alpha_i+\alpha_m.\]

Material gain, modal gain, power gain, and amplitude gain are different quantities. A Fabry–Perot amplifier formula requires a round-trip gain strictly below one; it cannot be used as a stable unsaturated amplifier above the lasing threshold.

Useful conversions are \(1\,\mu{\rm m}=10^{-4}\,{\rm cm}\), \(1\,{\rm \mathring A}=10^{-10}\,{\rm m}\), \(N[{\rm m}^{-3}]=10^6N[{\rm cm}^{-3}]\), and

\[E_\gamma[{\rm eV}]=\frac{1.239841984}{\lambda_0[\mu{\rm m}]}, \qquad {\cal R}=\eta_q\frac{e\lambda_0}{hc}.\]

An external differential quantum efficiency specifies a slope above threshold; it does not, by itself, specify the total output power at a given current without a threshold current or an intercept.