Coupling of a laser diode to a quadric interface lensed monomode elliptic-core step index fiber: efficiency computation with the ABCD matrix

Abstract

We report what to our knowledge is the first theoretical prediction of the coupling efficiency of a laser diode emitting two wavelengths either 1.3 or 1.5 μm, to three elliptic core step index fibers with different aspect ratios via quadric interface microlens of three different focal lengths on the fiber tip. Here, quadric interface microlens contains elliptical (hemispherical included), hyperbolic and parabolic microlenses. Instead of considering special ABCD matrix for individual microlens, a simple, accurate, and popular unified transfer ABCD matrix for refraction of quadric microlens under paraxial approximation is utilized to analyze the theoretical coupling efficiency, based on Gaussian beam approximation. The coupling efficiency can reach as high as 98.63 % with optimizing structure parameters of specific lensed fiber. The analysis can give some guidance for fabricating quadric interface lensed elliptic core fiber which are mainly useful in polarisation-maintaining fiber optic sensors and coherent fiber optic communication systems.

Appendix

The relation between input and output parameters (q ₁,q ₂) of the light beam is given by

$$ {q}_2=\frac{A{q}_1+B}{C{q}_1+D} $$

(A1)

where

$$ \frac{1}{q_{1,2}}=\frac{1}{R_{1,2}}-\frac{i{\lambda}_0}{\pi {w}_{1,2}^2{\mu}_{1,2}} $$

(A2)

with symbols having their usual meanings as already described.

The ray matrix M for the QML on the tip of the fiber is given by [24]

$$ \begin{array}{l}M=\left(\begin{array}{l}A\kern0.36em \\ {}C\end{array}\right.\left.\begin{array}{l}B\\ {}D\end{array}\right)\\ {}M=\left(\begin{array}{l}1\\ {}0\end{array}\right.\kern0.36em \left.\begin{array}{l}d\\ {}1\end{array}\right)\kern0.48em \left(\begin{array}{l}1\kern2.64em 0\\ {}\frac{1-\mu }{\mu P}\kern1.2em \frac{1}{\mu}\kern0.96em \end{array}\right)\kern0.6em \left(\begin{array}{l}1\kern0.6em \\ {}0\kern0.84em \end{array}\right.\left.\begin{array}{l}L\\ {}1\end{array}\right)\end{array} $$

(A3)

where

$$ A=1+\frac{d\left(1-\mu \right)}{\mu P} $$

(A4a)

$$ B=L+\frac{\left(1-\mu \right)Ld}{\mu P}+\frac{d}{\mu } $$

(A4b)

$$ C=\frac{1-\mu }{\mu P} $$

(A4c)

$$ D=\frac{1}{\mu }+\frac{\left(1-\mu \right)L}{\mu P} $$

(A4d)

where P is the structure parameter of the quadric interface lensed fiber, and L is the working distance which is also the distance of the LD from the microlens.

Again, the refractive index of the material of the microlens with respect to the incident medium is represented by $\mu \left(=\raisebox{1ex}{${\mu}_2$}\!\left/ \!\raisebox{-1ex}{${\mu}_1$}\right.\right).$ The transformed beam spot sizes and radii of curvature in the X and Y directions are found by using Eqs. (A4a-A4d) in Eqs. (A1) and (A2) and can be expressed as

$$ {w}_{2x,2y}^2=\frac{A_1^2{w}_{1x,1y}^2+\frac{\left({\lambda}_1^2{B}^2\right)}{w_{1x,1y}^2}}{\mu \left({A}_1D-B{C}_1\right)} $$

(A5)

$$ \frac{1}{R_{2x,2y}}=\frac{A_1{C}_1{w}_{1x,1y}^2+\frac{\left({\lambda}_1^2BD\right)}{w_{1x,1y}^2}}{A_1^2{w}_{1x,1y}^2+\frac{\left({\lambda}_1^2{B}^2\right)}{w_{1x,1y}^2}} $$

(A6)

where

$$ {\lambda}_1=\frac{\lambda }{\pi },\lambda =\frac{\lambda_0}{\mu_1}, {A}_1=A+\frac{B}{R_1}\kern0.96em and\;{C}_1=C+\frac{D}{R_1} $$

(A7)

In plane wavefront model, the radius of curvature R ₁ of the wavefront from the laser facet → ∞. This leads to A ₁ = A and C ₁ = C.