Coupling of a laser diode to single mode circular core graded index fiber via parabolic microlens on the fiber tip and identification of the suitable refractive index profile with consideration for possible misalignments

Abstract

We present a theoretical investigation of coupling optics involving a laser diode and circular core graded index single mode fiber via a parabolic microlens on the fiber tip to predict the nature of suitable refractive index profile for the first time in connection with optimum coupling. The study is carried out in absence and presence of possible transverse and angular misalignments. By employing Gaussian field distributions for both the source and the fiber and also ABCD matrix for parabolic microlens under paraxial approximation, we formulate analytical expressions for the concerned coupling efficiencies. The investigations are performed for two different light-emitting wavelengths of 1.3 μm and 1.5 μm for such fibers with different refractive index profile exponents. Further, it is observed that out of the studied refractive index profiles, triangular index profile having the dispersion-shifted merit comes out to be the most suitable profile to couple laser diode to such abovementioned fiber for two wavelengths of practical interest. The analysis should find use in ongoing investigations for optimum launch optics for the design of parabolic microlens either directly on the circular core graded index single mode fiber tip or such fiber attached to single mode fiber to achieve long working distance.

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 PML on the fiber tip is given by [26]

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

(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 focal parameter of the parabola, and L is the working distance which is also the distance of the LD from the PML.

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

$$ \begin{array}{cc}\hfill {\lambda}_1=\frac{\lambda }{\pi },\;\lambda =\frac{\lambda_0}{\mu_1},\;{A}_1=A+\frac{B}{R_1}\hfill & \hfill \mathrm{and}\hfill \end{array}\;{C}_1=C+\frac{D}{R_1} $$

(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.