Laser diode to circular core graded index single mode fiber excitation via quadric interface microlens on the fiber tip and identification of the suitable refractive index profile for maximum coupling efficiency with optimization of structure parameter

Abstract

We report what to our knowledge is the first theoretical prediction of the optimum coupling efficiency of a laser diode emitting two wavelengths either 1.3 or 1.5 µm to a series of circular core graded index single mode fibers with different profile exponents via quadric interface microlens of four different focal lengths on the fiber tip to predict the nature of suitable refractive index profile. Instead of considering special ABCD matrix for individual microlens like hyperbolic, parabolic and elliptical (hemispherical included) one, 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. Further, it is observed that out of the studied refractive index profiles, the step index profile comes out to be the most suitable profile to couple laser diode to circular core graded index single mode fiber for both wavelengths of practical interest. Moreover, the coupling efficiency can reach nearly 100 % with optimizing structure parameters of specific lensed fiber in both cases. The analysis should find use in ongoing investigations for optimum launch optics for the design of quadric interface microlens either directly on the circular core graded index single mode fiber tip or such fiber attached to single mode fiber.

Appendix

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

$$q_{2} = \frac{{Aq_{1} + B}}{{Cq_{1} + D}}$$

(10)

where

$$\frac{1}{{q_{1,2} }} = \frac{1}{{R_{1,2} }} - \frac{{i\lambda_{0} }}{{\pi w_{1,2}^{2} \mu_{1,2} }}$$

(11)

with symbols having their usual meanings as already described.

The ray matrix M for the QIML on the tip of the fiber is given by [28–30]

$$\begin{aligned} & M = \left( {\begin{array}{*{20}c} A & B \\ C & D \\ \end{array} } \right) \\ & M = \left( {\begin{array}{*{20}c} 1 & d \\ 0 & 1 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} 1 & 0 \\ {\frac{1 - \mu }{\mu P}} & {\frac{1}{\mu }} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} 1 & L \\ 0 & 1 \\ \end{array} } \right) \\ \end{aligned}$$

(12)

where

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

(13a)

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

(13b)

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

(13c)

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

(13d)

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 $\mu ( = {\raise0.7ex\hbox{${\mu_{2} }$} \!\mathord{\left/ {\vphantom {{\mu_{2} } {\mu_{1} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\mu_{1} }$}}).$ The transformed beam spot sizes and radii of curvature in the X and Y directions are found by using Eqs. (13a–13d) in Eqs. (10) and (11) and can be expressed as

$$w_{2x,2y}^{2} = \frac{{A_{1}^{2} w_{1x,1y}^{2} + \frac{{\left( {\lambda_{1}^{2} B^{2} } \right)}}{{w_{1x,1y}^{2} }}}}{{\mu (A_{1} D - BC_{1} )}}$$

(14)

$$\frac{1}{{R_{2x,2y} }} = \frac{{A_{1} C_{1} w_{1x,1y}^{2} + \frac{{\left( {\lambda_{1}^{2} BD} \right)}}{{w_{1x,1y}^{2} }}}}{{A_{1}^{2} w_{1x,1y}^{2} + \frac{{\left( {\lambda_{1}^{2} B^{2} } \right)}}{{w_{1x,1y}^{2} }}}}$$

(15)

where

$$\lambda_{1} = \frac{\lambda }{\pi },\quad \lambda = \frac{{\lambda_{0} }}{{\mu_{1} }},\quad A_{1} = A + \frac{B}{{R_{1} }}\;{\text{and}}\;C_{1} = C + \frac{D}{{R_{1} }}.$$

(16)

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.