Optics tab

Transfer Matrix Methods for Optical stratum

The confined optical mode in a 1D waveguide can be analytically solved using transfer matrix methods, cand calculating the root of the modal dispersion function \(\chi_M\)

\[\begin{split}&\begin{pmatrix} E_z(0) \\ H_x(0) \end{pmatrix} = \begin{pmatrix} \cos\alpha L & -i\gamma\sin\alpha L \\ -i\gamma^{-1}\sin\alpha L & \cos\alpha L \end{pmatrix} \begin{pmatrix} E_z (L) \\ H_x (L) \end{pmatrix} \\ &\alpha \equiv \sqrt{n^2k^2-\beta^2} \qquad \gamma\equiv \frac{\alpha}{kn^2} \sqrt{\frac{\mu_0}{\epsilon_0}}\\ &M_n \equiv \begin{pmatrix} \cos\alpha_n L_n & -i\gamma_n\sin\alpha_n L_n \\ -i\gamma_n^{-1}\sin\alpha_n L_n & \cos\alpha_n L_n \end{pmatrix} \qquad M \equiv M_1M_2\cdots M_N \\ &\chi_M \equiv M_{11}\gamma_s + M_{12} + M_{21}\gamma_s\gamma_0 + M_{22}\gamma_0\end{split}\]

[CH84].

Although generally it depends on effective refractive index as a complex function, it’s analytically within the domain of interest, therefore Newton’s method is applicable.

The roots of \(\chi_M(\beta)=0\) is the effective refractive index of guided modes. In the software we only capture the the mode with largest real part of \(\beta\), i.e. the foundmental mode. The imaginary part of \(\beta\) yeilds the waveguide loss

\[\alpha_w = \frac{4\pi}{\lambda}\mathrm{Imag}[\beta]\]

Confinement factor

There are multiple different definitions of confinement factor in different literatures, for example [JK14] and [YY06]. Here we use the following [LG20]:

\[\Gamma = \frac{\beta\int_{\text{AR}}n_z E_z^2\mathrm d y} {\int n_z^2E_z^2\mathrm d z}\]

Mirror loss

The mirror loss in the software can be chosen from cleaved surface ( refractivity \(R = |(\beta - 1)/(\beta + 1)|^2\)), perfect high-refraction coating (perfect HR, \(R = 1\)), perfect anti-reflection coating (perfect AR, \(R=10^{-1}\)) and customized refractivity.

The mirror refractivity leads to a effective mirror lose per waveguide length \(\alpha_M = -\ln(R_1 R_2)/2L\) where \(L\) is the waveguide length.

Threshold current

The threshold current is calculated assuming \(\eta_{\text{inj}} = 1\).

\[J_{th} = \frac{0.1\hbar c\epsilon_0 enL_p\alpha} {\eta_{\text{inj}} \,\text{FoM}}\]

See Optical gain and threshold current for detail.

Effective medium theory for QC layers

The QC layers have superlattice with width much less than wavelength, which is when efficiency medium theory works best. The effective refractivity index of the active region is not isotropic, for TM mode that we are interested in [LG20]:

\[n = \left\langle\frac{1}{n^2}\right\rangle^{-1/2}\]

[CH84]

John Chilwell and Ian Hodgkinson. Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides. J. Opt. Soc. Am. A, 1(7):742–753, Jul 1984. URL: http://josaa.osa.org/abstract.cfm?URI=josaa-1-7-742, doi:10.1364/JOSAA.1.000742.

[JK14]

Christian Jirauschek and Tillmann Kubis. Modeling techniques for quantum cascade lasers. Applied Physics Reviews, 1(1):011307, 2014. URL: https://doi.org/10.1063/1.4863665, arXiv:https://doi.org/10.1063/1.4863665, doi:10.1063/1.4863665.

[LG20] (1,2)

Ming Lyu and Claire Gmachl. Correction to the effective refractive index and the confinement factor in waveguide modeling for quantum cascade lasers. 2020. arXiv:arXiv:2007.03503.

[YY06]

Amnon Yariv and Pochi Yeh. Photonics: optical electronics in modern communications (the oxford series in electrical and computer engineering). Oxford University Press, Inc., 2006.